





■ 



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WORKS OF 
PROF. W. WOOLSEY JOHNSON 

PUBLISHED BY 

JOHN WILEY & SONS. 

An Elementary Treatise on the Integral Calculus. 

Founded on the Method of Rates. Small 8vo, 
ix + 234 pages, 36 figures. Cloth, $1.50. 

The Theory of Errors and the Method of Least 
Squares. 

i2mo, x-J- 172 pages. Cloth, $1.50. 
Curve Tracing in Cartesian Coordinates. 

i2mo, vi4-86 pages, 54 figures. Cloth, $1.00. 

Differential Equations. 

A Treatise on Ordinary and Partial Differential 
Equations. Small 8vo, xii + 368 pages. Cloth, 
$3.50. 

Theoretical Mechanics. 

An Elementary Treatise. i2mo, xv + 434 pages, 
115 figures. Cloth, $3.00, net. 

An Elementary Treatise on the Differential Cal- 
culus. 

Founded on the Method of Rates. Small 8vo, 
xiv -f- 404 pages, 70 figures. Cloth, $3.00. 



AN 
ELEMENTARY TREATISE 



ON THE 



DIFFERENTIAL CALCULUS 



FOUNDED ON THE 



METHOD OF RATES 



BY 



WILLIAM WOOLSEY JOHNSON 

Professor of Mathematics at the United States Naval Academy 
Annapolis, Maryland 



FIRST EDITION 
FIRST THOUSAND 



NEW YORK 
JOHN WILEY & SONS 

London: CHAPMAN & HALL, Limited 

1904 



<RA304 



T 



LIBRARY of CONGRESSJ 
Two Copies Received 

NOV 17 1904 

_, Copyrignt tntry 

cuss a, xXc. No; 
/of //<? 

COPY B. 



Copyright, 1904, 

BY 

W. WOOLSEY JOHNSON. 



ROBERT DRUMMOND, PRINTER, NEW YORK. 



<k 



PREFACE. 



In the year 1879, I published, in connection with the late 
Professor J. M. Rice, an Elementary Treatise on the Differential 
Calculus, founded on the Method of Rates or Fluxions. In the 
present work, the text is essentially new, and the order of the 
subjects will be found to differ in many points from that of the 
older book. I still employ the Newtonian conception of rates, 
and the functional method of obtaining the derivative, notably 
in the case of the logarithmic function, page 49 ; but the method 
is more closely connected with that of limits, and an independent 
basis for the algebraic functions is established in Art. 31. 

The differential of a variable not a linear function of the 
time is treated as an hypothetical increment corresponding to 
an assumed increment or differential of time, dt, page 18. It 
is assumed that there exists at any instant a definite measure 
of a rate, although it may be variable. In other words, the vari- 
ables dealt with being assumed continuous, their rates are also 
assumed to admit of a continuously varying measure. In velocity, 
the graphic form of rate, these hypothetical increments receive 
a beautiful realization in the use of Att wood's machine, see Art. 
21. The variable t (when it stands for elapsing time) is thus 
taken as the natural independent variable incapable of a non- 
uniform rate simply because, from its fundamental character, 
it is the standard to which all other variables are referred when we 
speak of their rates of variation. 



IV PRE FA CE. 

With this interpretation of dx, its ratio to dt is independent 
of the value of dt, and is the measure of the rate of x when a 
definite function of /. In the present work, the ratio of actual 
increments is at once shown to have the rate as its limit, and 
the evanescent quantity e expressing the difference is used in a 
general formula, Art. 26, for the non-linear function of t. From 
this formula the differential of the product, Art. 31, and thence 
those of the algebraic functions are readily deduced. 

When x is an independent variable, the derivative of a func- 
tion y is the measure of its relative rate, and the differentials 
become simultaneous hypothetical increments. The indepen- 
dence of their ratio upon their actual magnitudes is thus identified 
with the existence of a definite tangent to the curve which is 
the graph of the function. 

This notion leads naturally, in the case of the circular func- 
tions, to a mode of differentiation founded directly upon the geo- 
metrical definitions by which they are first introduced to the 
student, and not upon analytical properties subsequently de- 
duced. 

The advantage of giving to dx and dy separate meanings, 
and finite magnitudes no longer requiring continuous reference 
to the limit, is apparent in the mode of statement of the formulae 
of differentiation and the consequent treatment of the function 
of a function, Art. 48. 

Chapter VI presents a full treatment of the subject of de- 
velopment in power- series. In the text and answers to the 
examples will be found the developments of nearly all the 
elementary functions, including the direct and inverse hyperbolic 
functions. 

In the pages devoted to curves and curve tracing, the more 
essential algebraic methods as well as those peculiar to the 
Differential Calculus are exemplified. These pages will be found 
to contain a fairly full treatment of the higher plane curves 



PRE FA CE. V 

most commonly met with. Numerous entries in the alpha- 
betical index are given to facilitate reference to the results 
established. 

The topics have been so arranged that sections X., XII., 
XV., XVIII., XX., XXI., and XXXIII. can be omitted to 
form a shorter course to be supplemented by selections from 
the applications to curves and by the first section of the final 
chapter. 

W. Woolsey Johnson. 

June, 1904. 



CONTENTS. 



CHAPTER I. 

Functions, Rates and Derivatives. 
I. 

PAGE 

Continuous variables i 

Functions 2 

Implicit functions 3 

The graph of a function 4 

Inverse functions 4 

Continuity of a function 5 

Increasing and decreasing functions 8 

The linear function 9 

Functional equations 10 

Examples I n 

II. 

Rates of variation 14 

Constant rates 15 

Variable velocities 16 

Variables with rates not uniform 17 

Differentials 18 

Formula for the non-linear function of t 19 

The differential of the sum of several variables 20 

The differential of a term having a constant coefficient 21 

The differential of the product of two variables 22 

Examples II 23 

III. 

Rate of a function of an independent variable 25 

The derivative 26 

Graphic representation of the derivative 27 

Sign of the derivative 28 

Limit of the ratio of differences „ 29 

Examples III 30 



Vlll CONTENTS. 



CHAPTER II. 

Formula and Methods of Differentiation. 
IV. 

PAGE 

Character of the required formulae 32 

Differentiation of the square 32 

The differential of the product 35 

Differentiation of an inverse function 35 

Differentiation of a function of a function 36 

The rates of geometrical variables 37 

Examples IV =, 39 

V. 

The differential of the reciprocal 41 

The differential of the quotient of two variables 42 

Differentiation of x n 43 

Examples V 45 

VI. 

Differentiation of the logarithmic function. . . . 49 

Napierian logarithms 52 

The logarithmic curve 53 

Logarithmic differentiation , . . 54 

Differentiation of exponential functions 56 

The exponential curve 57 

Examples VI 58 

VII. 

Trigonometric or circular {unctions 61 

Differentiation of the sine and cosine 62 

Differentiation of the tangent and cotangent. . 63 

Differentiation of the secant and cosecant 64 

The logarithmic trigonometric functions 66 

Examples VII 66 

VIII. 

The inverse trigonometric functions 70 

Differentiation of the inverse sine 72 

Differentiation of the inverse tangent 73 

Differentiation of the inverse secant 74 

Homogeneous forms of the formulae 75 

Examples VIII. . , ,..,,..,... 76 



CONTENTS. IX 



IX. 

PAGE 

Recapitulation of formulae 78 

Differential of a function of two variables 80 

The derivative of implicit functions 82 

Examples IX 83 

Miscellaneous examples. . . 85 



CHAPTER III. 

Successive Derivatives. 

X. 

Velocity and acceleration 88 

Component velocities and accelerations 91 

Examples X 92 

XL 

Successive derivatives of a function 95 

Geometrical meaning of the second derivative 96 

Higher derivatives of implicit functions 98 

Examples XI 100 

XII. 

Expressions for the nth derivative 103 

Leibnitz' theorem 106 

Examples XII 109 



CHAPTER IV. 

Maxima and Minima. 

XIII. 

Characteristics of a maximum value 112 

Maxima and minima of continuous functions 113 

Maxima and minima of geometrical magnitudes 114 

Examples XIII 118 

XIV. 

Discrimination between maxima and minima 122 

Alternation of maxima and minima 124 

Employment of a substituted function 126 

Examples XIV 127 



X CONTENTS. 



XV. 

PAGE 

Implicit functions 129 

Maximum and minimum abscissae 132 

Infinite values of the derivative 132 

Functions of two variables 134 

Critical points on a surface 135 

Examples XV 138. 



CHAPTER V. 

Evaluation of Indeterminate Forms. 

XVI. 

Indeterminate or illusory forms 141 

Evaluation by differentiation 143, 

Functions which vanish with x 147 

Examples XVI 148 



XVII. 



GO 



The form %- 151 



CO 



Differential formula for the form ~ 153: 

The form 000 *55 

Limiting values of discontinuous functions 156" 

The form 00—00 .* . . 158 

Examples XVII *5 8 

XVIII. 

Functions whose logarithms take the form co .0 161 

Indeterminate forms of functions of two variables 163 

Application to the derivative of an implicit function 165 

Examples XVIII l6& 



CHAPTER VI. 

Development of Functions in Series. 

XIX. 

Series in ascending powers of x I 7 2 

Convergent and divergent series *73 

Differentiation of a series • • • • I 77 

Maclaurin's theorem r 79 



CONTENTS. XI 



Computation of e 181 

The binomial theorem 182 

Calculation of the development of tan x 184 

Examples XIX 184 

XX. 

Taylor's theorem 187 

Expressions for the remainder ,. 190 

Computation by numerical series 193 

Logarithmic series 194 

Computation of log e 2 and log e io 196 

Application to maxima and minima 197 

Evaluation of vanishing fractions by development 198 

Examples XX 200 

XXL 

The general term of the development. 201 

Employment of differential equations 202 

Computation of k 207 

Non-linear differential equations 210 

Examples XXI 212 

XXII. 

Even and uneven functions 214 

Hyperbolic functions 216 

Functions of pure imaginary quantities 218 

Complex quantities 221 

Conjugate complex quantities 222 

Functions of the complex quantity 222 

De Moivre's theorem 223 

Roots of unity 226 

Quadratic factors of certain algebraic expressions 227 

Examples XXII 228 

XXIII. 

The sine and cosine as continued products 232 

Bernoulli's numbers 237 

The development of (f> cot 240 

Examples XXIII 241 



XI 1 CONTENTS. 



CHAPTER VII. 

Application to Plane Curves. 
XXIV. 

PAGE 

Tangent and normal at a given point 247 

Tangent and normal at the origin 249 

Curves touching one of the axes at the origin 249 

The parabola of the wth degree 251 

Subtangents and subnormals 253 

The perpendicular from the origin upon the tangent 255 

Curve tracing 255 

Points of inflexion 258 

Examples XXIV 259 

XXV. 

Points at infinity 262 

The equation of the asymptote 264 

Tracing of curves with infinite branches 266 

Maximum and minimum coordinates 268 

Parabolic branches 269- 

Case in which one of the axes is an asymptote 270 

Examples XXV 272 

XXVI. 

Coordinates expressed in terms of a third variable 274 

The cycloid 275 

The prolate and curtate cycloids, or trochoids 278 

The epicycloid and the epitrochoid 278 

The hypocycloid and the hypotrochoid 280 

Epicyclics 281 

Double mode of generation 282 

Algebraic forms of the equations 284 

The four-cuspel hypocycloid, or astroid 286 

Employment of m as an auxiliary variable 286 

Examples XXVI 289 

XXVII. 

Polar equations 291 

Polar subtangents and subnormals 293 

The perpendicular from the pole upon the tangent 293 

Critical points 294 



CONTENTS. XI 11 



PAGE 

Zero values of r 295 

The lemniscate 296 

Polar equations involving only trigonometric functions of 6 297 

The limacon of Pascal 298 

The dygogram 300 

The cardioid 302 

Infinite values of r 303 

The perpendicular from the pole upon the asymptote 304 

Points of inflexion *. . 305 

Spirals 307 

The asymptotic circle 307 

The spiral of Archimedes 309 

The reciprocal spiral 309 

The logarithmic or equiangular spiral 310 

Examples XXVII 311 

XXVIII. 

The measure of curvature 314 

The radius and circle of curvature 315 

The radius of curvature where the curve is parallel to one of the axes 317 

The locus of the centre of curvature, or evolute 319 

The radius of curvature in rectangular coordinates 321 

The equation of the evolute 324 

The evolute of the cycloid 327 

The radius of curvature in polar coordinates 328 

Relations between p, p, and x 330 

Intrinsic equations 331 

The intrinsic equation and radius of curvature of the evolute 333 

Evolute of the catenary % 334 

Involutes and parallel curves 334 

Involute of the circle 337 

Examples XXVIII 337 

XXIX. 

System of curves 344 

Pencils of curves 345 

Envelopes 346 

Equations of the second degree in a 348 

General method 350 

Two variable parameters 352 

The evolute regarded as an envelope 353 

Envelopes of straight-line systems 354 

Examples XXIX 357 



XIV CONTENTS. 



CHAPTER VIII. 

Functions of Two or More Variables. 

XXX. 

PAGE 

Partial differentials 362 

Partial derivatives 364 

Geometrical representation of partial derivatives 365 

Higher derivatives 367 

Commutative character of differentiation 370 

Commutative and distributive operations 372 

Symbolic identities 375 

Euler's theorems concerning homogeneous functions 376 

Examples XXX 378 

XXXI. 

Change' of the independent variable 380 

Transformations involving partial derivatives 382 

Transformations of partial derivatives of the second order 384 

Transformations of p 2 387 

Symbolic and extended forms of Taylor's theorem 388 

Lagrange's theorem 389 

Examples XXXI 395 

Index 401 



THE 

DIFFERENTIAL CALCULUS. 



CHAPTER I. 

Functions, Rates and Derivatives. 



I. 

Continuous Variables. 

I. A MAGNITUDE of any sort which can be measured by 
a unit of its own kind is represented by the number of such 
units or parts of a unit which it contains. This number is 
called its numerical measure. If the magnitude is capable of 
indefinite subdivision, the corresponding numerical measure 
is conceived of as capable of an unlimited number of values. 
Such a numerical value, or " number," may be regarded as 
passing gradually from one fixed value to another in a certain 
interval of time, in such a manner as to assume during the 
interval every intermediate value ; it is then called a continuous 
variable. For example, if a point P is moving along a fixed 
straight line, its distance OP from a fixed point taken on 
the line is a continuous variable. 



2 FUNCTIONS, RATES AND DERIVATIVES. ' [Art. I. 

The variables with which we have to deal in the Differ- 
ential Calculus are always abstract quantities, or numerical 
values independent of any particular kind of unit, but it is 
frequently convenient to represent them graphically by linear 
distances. 

Functions. 

2. A quantity which depends for its value upon another 
quantity is said to be a function of it. A function of a con- 
tinuous variable will itself be a continuous variable, and the 
variable upon which it depends is in distinction called the 
independent variable. Denoting the independent variable by 
x, an algebraic or trigonometric expression containing x is 
generally a function of x, thus x 2 is a function of x\ but the 
expressions x°, x 2 -f- (a — x)(a -[- x), (tan x -f- cot ;tr)sin 2x, are 
not functions of x, since each admits of expression in a form 
which does not contain x. 

3. In treating of the general properties of functions, it is 
necessary to have a symbol which may denote any function 
of x. The symbol usually employed for this purpose is 
f(x), and when several functions occur in the same investi- 
gation, such symbols as Fix), F\x), (p(x), etc., are used, the 
enclosed letter in each case being the independent variable. 
Supposing the meaning of the functional symbol fin a given 
case to be denned, the value of the function corresponding to 
a particular value of x is expressed by substituting this value 
for x in the symbol f{x). Thus, if we are given 

u A x ) = x* — 2X, (i) 

we have J v J 

/(!) = -!, /(2) = 4) /(-!)=!, /(0)=0. 

Here equation (i) defines the meaning of f, and then the 
other equations follow; but none of these last equations 
serve, either alone or in combination, to define/". 



§ I.] FUNCTIONS. 



Again, if we are given 

F{x) = log a x, 
where a denotes the base of a system of logarithms, we have 
i^(i) =o, F(o) — — oo, F(a) = I. 

4. If a quantity which is independent of the independent 
variable x occurs in the expression for a function, it is, in 
contradistinction, called a constant, even when it has not an 
absolute value, but is represented (like a in the example above) 
by a letter to which any value may be assigned. Thus, when 
if is regarded as a function of x and denoted byy(^r), a is 
called a constant. When it is desired to put in evidence the 
fact that such a quantity is susceptible of different values, it is 
called a variable independent of x; the function then depends 
upon two independent variables. In this case, both variables 
are enclosed between the marks of parenthesis. Thus we 
may write 

fix, a) = a x ; 

and, taking this equation to define / as a function of two 
independent variables, we have 

f(y,b) = b\ /(3, 2) = S, /(2, 3) = 9 . 

Implicit Functions. 

5. When an equation is given involving two variables, x 
and y, either variable may be regarded as independent and 
the other as a function of it. If x is the independent variable, 
and y is not directly expressed in terms of x, it is called an 
implicit function of x. Thus, if we have 

ax 2 — ?>axy -f- y 3 — a z =Q, 



FUNCTIONS, RATES AND DERIVATIVES. [Art. 5. 



y is an implicit function of x, or is said to be a function of x 
given in the implicit form. Again, x is by virtue of the 
equation an implicit function of y. But, if we solve the equa- 
tion for x, we obtain 



4 



a 



and in this form, x is, in contradistinction, called an explicit 
function of y. 

The Graph of a Function. 
6. When a function is given in the explicit form 

y=A*\ • ■ • • • • • (1) 

it is often useful to construct the curve of which (1) is the 
equation as a graphic illustration of the mode in which the 
function or ordinate varies with the change 
of the independent variable or abscissa. 
For this purpose, rectangular coordinates 
are always employed, and the curve is 
called the graph of the function. For exam- 
ple, Fig. 1 shows the curve whose equation 
is y = x 2 . This curve, which is a parabola, 
is therefore the graph of the function x 2 . 




Inverse Functions. 

7. If an equation between x and y be solved in each of 
the forms 

y =/(*) an d x = <p(y), 

each of the functions f and <fi is said to be the inverse function 
of the other. Thus, ifjj> = x 2 , x = ± \/y ; the function ' 'square- 



I.] INVERSE FUNCTIONS. 



root" is therefore the inverse of the function " square." The 
graph of the inverse of a given function is the same curve as 
that of the given function in another position. Thus, Fig. i 
is the graph of the function square-root if we regard y as the 
independent variable, and Fig. 2, p. 6, is the graph when x 
is the independent variable. The square-root is a two-valued 
function ; but, since we can distinguish between the two 
values by means of their signs, the symbol tfx may be re- 
garded as a one-valued function. 

In the case of the trigonometric functions, a peculiar no- 
tation for the inverse functions has been employed. Thus if 



x 



= sin 0. we write = sin x x. 



Symbols of this character are, in the Calculus, always taken 
to denote the arcual measures of the angles in question or 
ratio of the arc to the radius, of which the unit is called the 
radian. For example, because the sine of 30 is £, and its 
arcual measure \ir t we may write \n = sin -1 |-. The inverse 
trigonometric functions are all many-valued functions, and 
the mode of distinguishing between the different values will 
be considered later. 

Continuity of a Function. 

8. A one-valued function of a continuous variable is itself 
a continuous variable, at least for certain ranges of values of 
the independent variable. For example, the function x 2 is 
continuous for all values of x; in other words, if y = x 2 , as x 
passes from any one given finite value to any other, y passes 
gradually , and not by any sudden leaps, from its first to its 
last value. Accordingly, the graph of the function, that is 
the curve y =x z , is a continuous line from the point for which 



FUNCTIONS, RATES AND DERIVATIVES. [Art. 8. 



x has any given value a to that at which x = b, as illustrated 
by Fig. I. 

On the other hand, the function square-root is continuous 

only for positive values of the indepen- 
dent variable. Thus, in Fig. 2, which 
is the graph of y = ^/x, considered as 
a one-valued function of x, the curve 
stops abruptly at the point where x = o, 
since there are no real values of the 
function for negative values of x. But 
Fig. 2. the curve is continuous between the 

points for which x has any two values >o. In this case, the 
function is said to become imaginary for values <o. 

9. Again, there may be values of the independent vari- 
able x in approaching which the value of the function y in- 
creases without limit. For example, the function 




y = 



x —I 



has a large value when x is a little greater than 1, and in- 
creases without limit as x ap- 
proaches nearer to 1. The func- 
tion is, therefore, said to become 
infinite when x =1. This func- 
tion has a negative value for all 
values of ;r<i, and approaches 
— 00 when x approaches 1. It is 
said to be discontinuous for any 
range of values which includes the 
value #■= 1. Accordingly, the 
graph of the function, Fig. 3, con- Fig. 3. 

sists of two branches which do not form a continuous line, so 




§!•] 



CONTINUITY OF A FUNCTION. 



that we cannot pass continuously from a value of the func- 
tion for which x is algebraically less than unity to one for 
which x> I. 

10. As a further illustration, the function 



y — 



X -f- I 



X 



(I) 



is real for all positive values of x\ but, when x approaches 
zero, it becomes infinite. More- 
over it is imaginary for values of x 
between zero and — I. But from 
x — — i to x = — oo , the function 
is again real. Thus it is continu- 
ous from x = o to x = oo , and also 
from x = — I to # = — oo. Ac- 
cordingly the graph, shown in 
Fig. 4, consists of two branches 
which are discontinuous one with 
the other. 



Fig. 4. 
It will be noticed that the equation 



xy l = x-\~ 1, 



(2) 



found by rationalizing equation (1), represents also the graph 
of the other value (which is negative) of the two-valued func- 
tion given in the implicit form by equation (2). Again, 
the complete curve, consisting of three branches, forms the 
graph of the function inverse to that considered, namely 



x = 



jr — 1 

This last is a one-valued function, and suffers discontinuity 
fby becoming infinite) when y = 1, and also when y = — 1 . 



8 FUNCTIONS, RATES AND DERIVATIVES. [Art. II 



Increasing and Decreasing Functions. 

11. A function is said to be an increasing one when its 
value increases with the increase of the independent variable. 
Such a function necessarily decreases with the decrease of the 
independent variable. On the other hand, a function is a 
decreasing one when it decreases with the increase of the inde- 
pendent variable, and consequently increases with the de- 
crease of that variable. 

The same function may of course be an increasing one for 
a certain range of values of the independent variable and a de- 
creasing one for another range. Thus sin x is an increasing 
function while x passes from o to |-7r; it is then a decreasing 
one while x passes from \7t to § n , since for this range of values 
of x its value decreases from its greatest value -f I to its least 
value algebraically considered, namely — I. 

12. When the graph of the function is drawn, algebraic 
increase of the independent variable x means motion from 
left to right, and algebraic increase of y means motion upward. 
Hence, supposing a point to describe the graph moving 
toward the right, it will move upward if the function is 
an increasing one. In this case, the slope of the curve is said 
to be positive. In the opposite case, representing a decreas- 
ing function, the slope is negative. Thus in Fig. I, the graph 
of the function x* y the slope is positive and the function is 
an increasing one for all positive values of x. But the slope 
is negative and the function is a decreasing one for all nega- 
tive values of x, since y decreases as these values algebra- 
ically increase. 

Again, Fig. 2 shows that the positive value of the inverse 
of this function is an increasing function. Fig. 3 has a nega- 



§ L] THE LINEAR FUNCTION. 9 

tive slope throughout both branches, so that is always 

a decreasing function. In like manner, Fig. 4 shows that 
x + 1 



v 



is, when real, always a decreasing function of x. 
x 



The Linear Function. 




13. The simplest of all functions is the rational algebraic 
expression containing x only in the first degree. Its general 
form is 

y =z MX -j- b, 

where m and b are constants. The corresponding graph is 
therefore an oblique straight line, 
and for this reason the function is 
said to be of the linear form. The 
linear function is continuous for all 
values of x, and is throughout either 
an increasing one or a decreasing one, 
according as m is positive (as in 
Fig. 5) or negative. The slope of 
the graph is constant, and m (the tangent of the angle it 
makes with the axis of x, or ratio of the rise to the corre- 
sponding horizontal distance) is taken as its measure, and is 
called the gradient. 

14. In the case of all other functions, the graph has a 
variable slope. If the value of the function y can be ex- 
pressed either rationally or fractionally in terms of powers of 
x, or, in the case of implicit functions, if the relation between 
x and y is algebraic, so that the graph is an algebraic curve, 
the function is said to be an algebraic one. All other func- 
tions are called transcendental. 



10 FUNCTIONS, RATES AND DERIVATIVES. [Art. 1 5, 



Functional Equations. 

15. An equation involving an unknown function, that is 
to say, the values of such a function corresponding to differ- 
ent values of the variable upon which it depends, is called a 
functional equation. The values of the variable may either 
be connected or independent of one another. In either case, 
the equation expresses a property of the function. Thus 
f(x) = f{ — x) expresses a property of the function symbol- 
ized for the present by f. The information conveyed by this 
equation does not go far toward determining f y because the 
property is shared by a great variety of functions, such as 
cos^r, x* -\- x 2 , e* 2 , etc. 

On the other hand, such an equation as 

f{*y)=f{x)+f{y), 

where x and y are independent of one another, expresses a 
highly characteristic property of the function f. It will, in 
fact, be recognized as the characteristic property of the loga- 
rithmic function ; namely, that which expressed in words is 
the rule that "the sum of the logarithms of any two num- 
bers is the logarithm of their product." 

16. The solution of a functional equation consists in find- 
ing the most definite expression for the unknown function 
which will include all the functions which have the given 
property. When the equation contains two independent 
variables, like that above, the solution is effected if we can 
separate the variables, so that each occurs on one side only 
of the equation. For, by so doing, we must necessarily 
obtain an expression which has the same value for any two 
(and therefore for alt) values of the independent variable. 



§ I.] FUNCTIONAL EQUATIONS. II 

Such an expression must have a constant value, and this fact 
gives us an equation containing only one variable. 

For example, suppose that, with respect to an unknown 
function f it can be shown that 

xf{x) = Z f(B), (I) 

while x and z are independent, so that x may vary while z 
retains a fixed value. Let c be the value of the expression 
zfiz) when z has a given fixed value. Then, by virtue of 
equation (i), 

xf{%) = c (2) 

In this equation, c is a constant because it is independent of 
x ; hence it appears that, in this case, the expression xf(x) is 
not itself a function of x> because it has a value independent 
of x. It follows that 

is the most definite expression we can give to fix), unless 
some other known property of the function serves to deter- 
mine the constant c. 

Examples I. 

i. (a) For what value of n does x n cease to be a function of a:? 
i/3) For what values of x does it cease to be a function of n ? 

(a) When n = o. (/?) When x = i and when x = o. 

(d x\ CLX Jt 2 
i — —J = x -I , show that vis a function 
a -\- xj a -j- x 

of a, but not of x. 

3. Show that sin x tan \x -j- cos x is not a function of x. 

4. If y = x -j- 4/(1 -J- jv 2 ), show ih.3ity 2 —2Xjy is not a function of jr. 

5. If/"(.#) = .r 2 , find the value off(x -j- ^); of/(2^); of /(.r 2 ); 
.of/C**-*); of/(i); of/(i2); of/[/(.r)]. 



12 FUNCTIONS, RATES AND DERIVATIVES, [Ex. I. 

6. \if{x) =log (i +x), find the value of /(o); of/(i); of 
/(co); of /(-i). 

7. If/(#) = cos 6/, find the value of/(o); of f($7r); of/(£7r); 
of/(7r); of/(f7r). 

8. If ^(» = a x , give the value of F(a); ofi^(i); of F(o). Also 
show that in this case [i^(^-)] 2 = F(2x). 

9. Given xy — 2.x -\-y — n, show that y is not a function of x 
when n = 2. 

10. Given jy 2 — 2#y -)- x 2 = o, make jy an explicit function of x, 
and draw the graph of the function. y = a ± ^/(a 2 — ^). 

11. Given 1 + \og a y = 2 log a (^ -j- a )> make jy an explicit func- 
tion of x. (x -\- a) 2 

y=z - ■ - ' 

y a 

12. Given the equations : 

n -J- 1 = n (cos 2 #'-|- cos 6' cos 6 -f- cos 2 #), 
and n — 1 = n (sin 2 ^ -f sin B' sin B -f- sin 2 #); 

regarding 6 as the independent variable, determine 6' and n as explicit 

functions of B. or n . . 1 

B' — B ± i?t, and « = q= -r— s - 

sin C7 cos f7 

13. Given sin ~ 1 x — sin ~ l y = a, make y an explicit function of x. 

y = x cos a ± sin <r 4/(1 — jv 2 ). 

14. Given tan ~ 1 x -j- tan ~" ^ == tf, make jy an explicit function of x. 
Also show that x is the same function of y, and point out the cor- 
responding peculiarity of the graph. _ tan a— x 



1 -j- x tan a 

2,JC I 

it? If y — show that the inverse function is of the same 

D $X — 2' 

form. 

1 —I— x 
16. If y =f(x) = — — — , find z =f(y), and express z as a func- 



tion of x, that is to say, fmo\jy~(x). Also find ffffix) . 



1 

x 



17. Find the inverse of the function^ = log a [^r -f- 4/(1 + -X" 2 )] . 

x= \{a? - a-v). 



§ L] EXAMPLES. 



18. For what ranges of values of x is tan x a continuous function, 
and for what ranges an increasing function ? 

19. If both f "and denote increasing functions, and also if both 
denote decreasing functions, show that 0[_/(-X")] is an increasing 
function. Show also that the inverse of an increasing mnction is an 
increasing function. 

20. Show, by consideration of the graph, that a function which is 
continuous for all values of x can be infinite only when x is infinite. 

21. If a function continuous within a certain range is sometimes 
an increasing and sometimes a decreasing one, show that its inverse 
cannot be a one-valued function. 

22. State the peculiarity of the graph of a one-valued function 
having the property 

and show that, if the function is defined by _/"(.#) = (p{x 2 ), under 
this condition only can be a one-valued function. 

23. If f{x) is an unknown function having the property 

/(*) +/O0 =/(*>>), 

prove that 

Put y = 1. 

24. If fix) has the property 

A*+S)-=A*)+Ay)> 

prove that /"(o) = o. Also prove that the function has the property 

AP X ) =PA X )> 

in which p is a positive or negative integer. 

For positive integers, put y = x, 2X, ^x, etc. in the given equation; 
for negative integers, put y = — x. 

25. If f denotes the same function as in Example 24, prove that 



f{mx) = mf(x), 



14 FUNCTIONS, RATES AND DERIVATIVES. [ Ex. I. 



m denoting any fraction. 

P 
Put z = —x, p and q being integers, and apply the result of Ex. 24. 

26. Denoting by /"the same function as in the preceding examples, 
derive from the general property 

f(mx) — mf{x) 
the result 

and thence deduce the form of the function. See Art. 16. 

f(x) = ex* 

27. Given [0 (*)]*= [0(«)]*, and 0(i) = *, 
determine 0(#). 1 

0(^) = e x . 

28. Given 0(*) + 0(_>') = 0(-^J^), 
prove <fr(x m ) = mcp(x), 
and thence prove 0(-^) = £ log x. 

Use the methods of Examples 23, 24, <2«</ 25. 



II. 

Rates of Variation. 

17. In the Differential Calculus, quantities susceptible of 
continuous variation are treated of by means of the rates of 
their variation. Let a continuous variable be represented, as 
in Art. 1, by the distance OP of a point moving along a 
straight line from a fixed origin of distances taken on the 
line ; then the rate of increase of the variable is represented 
by the velocity of the moving point. If the line is horizon- 
tal, as in Fig. 6, distances to the right being, as usual, 
regarded as positive, a rate of increase is represented by 



§11.] CONSTANT RATES. I 5 

motion to the right, and this is taken as a positive rate. 
Accordingly, motion toward the left corresponds to a rate of 
algebraic decrease, or a negative rate. 

Denote the variable by x, and denote by t the time, as 
measured from some fixed instant 

taken as the origin of*time; then, in 1 - — ^ 7- 

this representation, we are in effect 

making x a function of /. Negative 

values of t correspond to positions occupied by P at instants 

before that chosen as the origin of time. 

Constant Rates. 

18. A variable is said to have a constant rate, when it 
receives equal increments in any equal intervals of time ; in 
other words, when the differences of the values corresponding 
to the beginnings and ends of any equal intervals are equal. 
Under these circumstances, the point P in the graphic illus- 
tration, Fig. 6, will have a uniform velocity. The incre- 
ments of the variable x mentioned above are now repre- 
sented by spaces described by P in intervals of time having 
some fixed magnitude. The definition of uniform motion or 
constant velocity requires that these spaces should be equal. It 
readily follows that the spaces passed over in any intervals of 
time are proportional to the intervals, and this may be taken 
as a more convenient definition of a constant velocity. 

19. The numerical measure adopted for a constant velocity 
is the number of units of space passed over in the unit of 
time (generally the second), and accordingly the measure of 
a constant rate for any variable is the increment received in a 
unit of time. Let k denote this increment (or, in the illus- 
tration, the space described in a unit of time), then the 



1 6 FUNCTIONS, RATES AND DERIVATIVES. [Art. 1 9. 

increment received in / units of time by a variable x having 
the constant rate k will be kt. Hence, if a denotes the value 
of x at the time / = o, the value of x at the end of / units of 
time will be 

x = a -f- kt (1) 

Thus a uniformly varying quantity is a linear function of the 
time elapsed since a given instant, that is, of the time as 
measured from a fixed origin of time. 

Conversely, whenever a variable x is a linear function of 
the time, it has a uniform rate ; and the coefficient of /, when 
x is expressed in the form (1), is the measure of this rate. 

Variable Velocities. 

20. If the velocity of a point be not uniform, its numeri- 
cal measure at any instant is the number of units of space 
which would have been described in a unit of tune, if the 
velocity had re7nained constant from and after the given 
instant. 

Thus, when we speak of a body as having at a given 
instant a velocity of 32 feet per second, we mean that, should 
the body continue to move during the whole of the next 
second with the same velocity which it had at the given 
instant, 32 feet would be described. The actual space de- 
scribed may be greater or less, in consequence of the change 
in velocity which takes place during the second ; it is, for 
instance, in the case of a falling body greater than the meas- 
ure of the velocity at the beginning of the second, because 
the velocity increases throughout the second. 

21. Attwood's machine for determining experimentally 
the velocities acquired by falling bodies furnishes an example 



§11.] VARIABLE RATES. 1 7 

of the practical application of the principle embodied in the 
above definition. 

This apparatus consists essentially of a thread passing 
over a fixed pulley, and sustaining equal weights one at each 
extremity, the pulley being so constructed as to offer the 
slightest possible resistance to turning. On one of the weights 
a small bar of metal is placed, which, destroying the equilib- 
rium, causes the weight to descend with an increasing velocity. 
To determine the value of this velocity at any point, a ring 
is so placed as to intercept the bar at that point, and allow 
the weight to pass. Thus, the sole cause of the variation of 
the velocity having been removed, the weight moves on uni- 
formly with the required velocity, and the space described 
during the next second becomes the measure of this velocity. 

Variables with Rates not Uniform. 

22. When a variable quantity x is represented, as in 
Fig. 6, by a distance measured upon a fixed straight line, 
the increment which it receives in a certain interval of time is 
represented by the space passed over by the moving point in 
that interval. Such an increment (or difference of values of x) 
is denoted by the symbol Ax, and the interval of time to which 
it corresponds is denoted by At. Then the characteristic of 
a variable which has not a constant rate is that Ax is not pro- 
portional to At, so that neither the increment received in 
a unit of time, nor the ratio of any simultaneous increments, 
furnishes a measure of the rate. Under these circumstances, 
the measure of the rate at a given instant is taken to be 
the same as that of the variable velocity which represents 
it ; namely, it is the increment which would have been received 
in a unit of time, if the rate had remained constant for the 
"whole of that interval. 



l8 FUNCTIONS, RATES AND DERIVATIVES. [Art. 23. 



Differentials. 

23. The increment which would be received by x in any 
chosen interval of time, on the hypothesis made above, is 
called a differential. Such an hypothetical increment is de- 
noted by dx; while the interval of time to which it corresponds 
is denoted by dt and is called the differential of time. The 
differential of time can be chosen of any magnitude; and, 
considering several values of dt, the corresponding values of 
dx will be proportional to them. It follows that the ratio 

dx 
dt 

will be the measure of the rate of x, no matter what the value 
of dt. 

24. If now we put dt = At we shall not, in general, have 
dx = Ax. This will be true only when x varies with a con- 
stant rate ; that is to say, when x is a linear function of t. 
Therefore, writing the equation 

Ax dx , 

57=^+' tO 

e is a quantity which in general will have a positive or nega- 
tive value. 

This quantity e is the difference between the rate of x 
and the ratio of certain actual corresponding increments of 
x and t. Since this difference is due solely to the fact that 
the rate of x changes during the interval At, e is a function 
of At which vanishes with At. 

25. With this understanding of the nature of e, equation (1) 
expresses the fact that the limiting value of the ratio of the 



§11.] FORMULA FOR THE NON-LINEAR FUNCTION. 19 

increments Ax and At, when At is diminished without limit ; , is 
equal to the measure of the rate of x* 

Formula for I he Noji- Linear Function of t. 

26. A variable x which has not a uniform rate is a non- 
linear function of t. Such a function may be expressed in a 
form which exhibits the difference between it and the linear 
function. For this purpose, we make use of the value which 
x has at any particular instant, and for simplicity take that 
instant as the origin of time. Denote by a the special value 
of x thus taken to correspond to / = o ; this may be called 
the initial value of x. Then, passing to any other correspond- 
ing values of x and t, we have 

t = At and x = a-\- Ax (2) 

Denote also the value which the rate of x has when t = o, that 
is, the initial value of the rate, by k. Now, since the rate is 
not constant, we have, by equation (1), Art. 24, 

Ax = (k 4- e) At ; 

whence, equation (2) becomes 

x = a + {k + e) /, . . . . . (3) 

where e is, by Art. 24, a function which vanishes when t = o. 

27. It follows that, when a given function of t is put in 
this form, k is the value which the coefficient of t assumes 
when we put t = o. Thus, if the coefficient of / is variable, 

* In accordance with this theorem, the hypothetical differences or differentials, 
corresponding to any particular simultaneous values of x and /, are quantities 
which (while they may have any magnitudes whatever) must always have that 
ratio which is the limiting value of the ratios of the actual increments, when 
made indefinitely small 



20 



FUNCTIONS, RATES AND DERIVATIVES. [Art. 2J, 



the rate is variable, and its value when t = o is obtained by 

putting t = o in the coefficient, 

dx 
We have seen in Art. 23 that -77 is the symbol for the 

rate of x. When this is variable, the value of it which cor- 



dx~ 



responds to a special value a of t is denoted by -=- 



dtj: 



Thus 



the theorem just proved is that, when x is put in the form 

x — a -f- {k -f- e)t y 
where e vanishes with /, 

dx~\ 



dt 



= k. 



The Differential of the Sum of Several Variables. 

28. Let x and y denote any two variables and k and k' 
their rates at any given instant. Then, taking this instant as 
the origin of time, their values may, by the preceding articles, 
be written 

x = a + {k + e)t, (1) 

y = b+(k'+ e')t (2) 

Hence their sum is 



x +y == (a + b) + (k + ¥ -f e -f /)/, 



(3) 



which is of the same form. Since e -j- e' vanishes ivhen 
t = o, the coefficient of t takes the value k -j- k' when t = O. 
Thus, by Art. 27, 



d(x + y) ~[ 
dt J, 



*^ k ~ dtA 



ay 
y dtA: 



§11.] THE DIFFERENTIAL OF THE SUM. 21 

The several rates in this equation are their values at the 
instant chosen as the origin of time. But, since any instant 
may be so chosen, we have proved that at all times 

d(x -j- y) dx dy 

~dt ~di + di' ^ 

that is, the rate of the sum of two variables is always equal to 
the sum of their rates. 

29. Multiplying by dt we have 

d{x-\-y) = dx-\-dy, (5) 

in which it is of course to be understood that all the differ- 
entials correspond to the same value of dt. 

This formula is readily extended to any number of varia- 
bles. Thus 

d(x-\- y -f- z -)-....) = dx-\- d(y -\- z -f- . . .) 

= dx -j- dy -f- dz -\- (A) 

that is, the differential of the sum of any number of variables 
is the sum of their differentials. Since a constant has no dif- 
ferential it appears that in differentiating a polynomial, con- 
stant terms do not affect the result. 

The Differential of a Term having a Constant 

Coefficient. 

30. Let the term be denoted by ntx, m denoting a con- 
stant. Putting, as in Art. 26, 

x = a -f- {k -\- e)t, (1) 

we have 

mx = ma -f- (mk -f- me)t (2) 



22 FUNCTIONS, RATES AND DERIVATIVES. [Art. 30. 

Since me vanishes when / = o, the coefficient of / assumes 
the value mk, and we have, by Art. 27, 



d(inx) 
dt J 



dx 

mk = m —r- 
dt 



Hence, at the instant chosen as the origin of time, the rate of 
mx is m times the rate of x. But, as this may be any instant, 
the same thing is true generally, and multiplying by dt, 

d(inx) = mdx. 

It therefore follows that the differential of a term having 
a constant coefficient is equal to the product of the differential of 
the variable factor by the constant coefficient. 

The constant m may have a negative value, and in par- 
ticular 

d{ — x) = — dx. 

The Differential of the Product of Two Variables. 

31. Let x and y be any two variables, and, selecting any 
instant as the origin of time, express them, as in Art. 26, by 

x = a -f- {k -\- e)t, 
y = b + (k'-\- e')t. 

Then their product is 

xy = ab + \bk + ak' + be + ae' + {k + e)(k' + e')f\t. 

Assuming the initial values and rates, a, b, k and k\ 
to be finite quantities, the terms be and ae' as well as 
{k -\- e){k' -f- e')t vanish with t ; hence the coefficient of t 
reduces when / = o to bk-\-ak', which is therefore, by Art. 
27, the measure of the rate at the instant t = o. Thus, at the 



§ II.] EXAMPLES. 23 

chosen instant, the rate of xy is the sum of the products of 
the values of each variable and the rate of the other. 

Since this is true at every instant, we have therefore in 
general 

d{xy) dy dx 
~df ~ X ~di +y ~d~r 

whence 

d(xy) = xdy -j- ydx. 

Examples II. 



2X X 

I. Find the differential of — and of 



3# m — 2 



2dx . dx 
and 



3# m — 2 



2. Find the differential of t— and of z—. 



dj 

'i m2 



dx . dx 
— and ^ . 



3. Find the differential of * + b +^ ~ 5)x . dx 



a + b' 

4. Find the differential of ^±4 and { b ^±A 

a -f- a(a -f- b) 

dx and K ** + dy) 
a -\- b a{a -j- b) 

5. Given ay -f- bx -f- 2cx -\- ab = o, to find -^. 

dx 

& __ b -f- 2C 

dx ' a 

6. Given _>> log a -f- x sin a —y cos a — ax -f tan a = o, to find 
dy 

dx* dy a — sin a 

dx log a — cos a 



24 FUNCTIONS, RATES AND DERIVATIVES. [Ex. II. 

7. Given ay cos 2 a — 2<5(i — sin a)x =. b{a — x cos 2 a), to find 
dy 

dx' dy___ b(i — sin a ) 

dx~~ a{\ -\- sin a) 

dy 

8. Given a 2 -\- 2(1 -(- cos a]y = (x ~\- y) sin 2 a, to find — . 

dy a 

— = tan^ — . 
dx 2 

o. Given \- -f- -I — = 1, to express ^!s in terms of dx and <f^. 

a c 

c c 

dz = dx dy, 

a 

10. The sides of a rectangle have the values 12 and 9 inches at a 
given instant, they are then increasing uniformly, the first at the rate 
of 2 inches per second and the second at the rate of 1 inch per 
second. At what rate per second is the area increasing ? 

30 square inches. 

11. How would the area be changing if the first side were 
decreasing, other things being as in Ex. 10 ? 

12. In each of the last two examples what will be the rate after 
the lapse of one second ? 

13. A man whose height is 6 feet walks directly away from a 
lamp-post at the rate of 3 miles an hour. At what rate is the extrem- 
ity of his shadow travelling, supposing the light to be 10 feet above 
the level pavement on which he is walking ? 

Draw a figure, and denote \ the variable distance of the man from the 
lamp-post by x, and the distance of the extremity of his shadow from the 
post by y. j-^ miles per hour. 

14. At what rate does the man's shadow (Ex. 13) increase in 
length ? 



§111.] RATE OF A FUNCTION. 2$ 

III. 

Rate of a Function of an Independent Variable. 

32. In many applications of the Calculus, the variables 
treated of are actual functions of the time, and therefore have 
definite rates, which being generally variable are themselves 
functions of the time. The spaces passed over by the falling 
body in Arts. 20 and 2 1 afford an illustration : the velocities as 
well as the spaces are definite functions of the time. 

In other applications, the variables concerned, although 

connected together, have no necessary connection with elapsing 

time. In these cases, one of the variables is arbitrarily 

chosen as the independent variable. Denoting it by x, it is 

dx 
assumed to have a rate — - of arbitrary and constant value. 

dt J 

dy 
Then, if y is a given, function of x, its rate ~ will depend for 

{Li/ 

its value not only upon the functional relation of y iox, but 

also upon the rate assumed for x. 

33. Now, since dy and dx in the symbols for the rates 
imply the same value of dt, we have 

rate of y dy 
rate of x dx 

that is to say, the ratio of the differentials expresses the rel- 
ative rate of y when the rate of the independent variable x 
is taken as the standard. Since the rate of x is assumed to 
be constant, the relative rate of y will, in general, be variable ; 
that is, it will be a function of the independent variable. 

dy 
But it is important to notice that this ratio ~ is not a 

dx 

function of dx; for the value of it which corresponds to a spe- 



26 FUNCTIONS, RATES AND DERIVATIVES. [Art. 33. 

cial value of x is absolutely determined by the rates of y and 
x, and yet the values of dy and dx can be changed by altering 
the value of dt. 

The Derivative, 
34-. It follows that, when 

y =/(*)> (1) 



we may put 



!='<•>• (2 > 



where f is a new function of x. The new function thus de- 
rived from the given function/ is called the derived function, 
or more commonly the derivative of the function/". 
Equation (2) may also be written in the form 

dy=f(x)dx, ...... (3) 

and, for this reason, the derivative f'{x) is also sometimes 
called the differential coefficient of y. 

It is only in the case of the linear function 

y = mx + b, 
which gives 

dy = mdxj 

that the differential coefficient is a constant and not a function 
of x. 

35. We are said to differentiate a function of a single 
independent variable, when we express its differential in the 
form (3). The first part of our work in the following chap- 
ters will be to obtain the formulae or rules for the differentia- 
tion of the simple functions in this form. The expression of the 

value of -j- in the form (2) is called taking the derivative of 

CLX 

y with respect to x. 



§ III.] GRAPHIC REPRESENTATION. 2J 

It is to be noticed that the result of this operation, of 

which the symbol is — -, is simpler than that of differentiation, 

dx 

of which the symbol is d. The reason is that dy is a function 

of the independent quantity dx* as well as of x; whereas the 

dx implied in the value of dy, equation (3), has been removed 

from the derivative by division. 

Graphic Representation of the Derivative. 

36. When the graph of the function y =f(pc) is drawn, as 
in Art. 6, the simultaneous values of x and y are the rectan- 
gular coordinates of a moving point which describes a continu- 
ous line. If the function is linear, this line is straight, and 
the moving point is said to have a constant direction. We 
have seen in Art. 34 that, in this case, the ratio of the rate of 
y to that of x is constant and equal to m, which, in the equa- 
tion y = mx-\- b, is the tangent of the angle which the straight 
line makes with the axis of x. 

37. For all other functions, a curve is described and the 
direction of the moving point is 
variable. Let the curve in Fig. 7 be 
the graph of the function y=f(x). 
Since we have assumed (Art. 32) the 
rate of x to be constant, the rate of y 
is now variable. When the moving 
point arrives at a particular position, 
as P in the diagram, let us suppose 
the rate of y to become constant with- Fig. 7. 

out suffering change in value, The moving point will then 




* Thus d is not a functional symbol, for dx is independent of x. We cannot 
however say that dy is independent of y, since it is a function of x upon which y 
depends, and therefore has different values for different values of y. 



28 FUNCTIONS, RATES AND DERIVATIVES. [Art. 37. 



describe a straight line PP\ and the constant direction of this 
line exhibits the direction of the moving point in the curve at 
the instant it passes the point P. 

This straight line passing through a given point of a curve 
and having the direction of the curve at that point is called 
the tangent to the curve at that point, which is called the 
point of contact. 

38. Let PP\ Fig. 7, be the space described in the time 
dt by the point considered in the preceding article, then the 
difference of abscissae PB represents dx, and the difference of 
ordinates BP' represents dy. The fact that the ratio of these 
differentials is independent of their magnitude is illustrated 
by the similar triangles in the figure. 

Let denote the inclination of the tangent line to the 
axis of x, then we have 

dy P'B 
s =_ = tan0 : 

Thus the trigonometric tangent of the inclination of the graph 
of y = f(x) is the graphic representation of the derivative of 
the function. 

It will be noticed that there are two values of 0, differing 
by 180 , according as we suppose the point to move in one 
or the other of the two opposite directions in the curve; but 
the value of tan is the same for these two values, since 
tan (0 -f- 180 ) = tan 0. 

Sign of the Derivative. 

39. By the definition given in Art. II, a function y = f(x) 
is an increasing one, when x and y increase together or de- 
crease together; in other words, when the rates of x and y 

have the same algebraic sign. In this case, the ratio ~~ 

dx 



§ III.] LIMIT OF THE RATIO OF DIFFERENCES. 29 

is positive. Thus fix) is an increasing function for all values 
of x which make the derivative f'(x) positive ; and, on the 
other hand, it is a decreasing function for all values which 
makef'(x) negative. Accordingly, when the graph is drawn, 
tan (p is taken as the gradient or measure of the variable slope 
of the curve which, as stated in Art. 12, is positive in the 
first case and negative in the second. 

Limit of the Ratio of Differences. 

40. Denoting actual increments, as in Art. 24, by At, Ax 
and Ay, we now have Ax = dx when At = dt, because x is 
assumed to have a uniform rate. But we have not Ay = dy. 
The difference is illustrated in Fig. 7, where the actual in- 
crement of the ordinate is Ay = BQ, terminating in the curve, 
while the hypothetical increment is dy = BP' , terminating in 
the tangent line. If, after the analogy of Art. 24, we put 

^1 --^1 -L e ( ) 

Ax dx 

e will be a quantity which vanishes with Ax. Thus when 
y = f(x), the derivative is the limiting value of the ratio of 
simultaneous increments or differences, when the absolute values 
of these increments are diminished without limit.* 

41. The quantity e in equation (1) is illustrated in Fig. 7 by 
the ratio of P ' Q to PB. Not only does P' Q vanish with PB 
or Ax, but its ratio to Ax vanishes. The first member of the 
equation is the trigonometric tangent of the inclination, that 
is to say, the slope of a secant line which cuts the graph in 

* The actual increments or differences are sometimes called finite differences in 
distinction from the small differences of which the limits only are considered in 
the Differential Calculus. The latter are then called infinitesimal differences, an 
infinitesimal being denned as a quantity having zero for its limit. 



30 FUNCTIONS, FATES AND DERIVATIVES. [Art. 41. 

the points P and Q, of which the abscissae are x and x -(- Ax. 

The ratio in the second member is the slope of the tangent 

at the former point. Thus the geometrical equivalent of 

the proposition stated above is that the tangent is the limiting 

position of the seca?vt line. 

dy 
The special value which the derivative — takes when x 

dyl 
has a special value a is denoted by -7- . Thus, if y = f(x) t 



Examples III. 

1. If a point moves in the straight line 2jy — *jx — 5 = o, so that 
its ordinate decreases at the rate of 3 units per second, at what rate 
is the point moving in the direction of the axis of x ? 

dx 6 

~di~ ~ 7' 

2. If a point starting from (o, b) moves so that the rates of its 
co-ordinates are k and k' ' , show that its path isy = mx -j- b> m being 

a 

equal to — . 

Express x andy in terms oft {Art. igi) and eliminate t. 

3. If a point moving in a curve passes through the point (5, 3) 
moving at equal rates upward and toward the left, find the value of 
dy' 



dx 



, also the equation of the tangent line to the curve at the given 

dy 
point. 



5 

dx_ 

4. If a point is moving in the straight line 

x cos a -\-y sin a-=p, 



— — i;y-{- x = 8. 



§ III.] EXAMPLES. 31 

its rate in the positive direction of the axis of x being / sin a, what 
is its rate of motion in the direction of the axis of^ ? 

— / cos a. 

5. Given ay sin a — ax -f- ax cos a — b 2 sec a = o; show that 
is constant and equal to \a. 

6. \if{x) = tan x, show thaty^.*) must always be positive. 

dy 

7. Show, by tracing the graph, that if y = x 3 — can never be 

negative. 

8. Given the property of the parabola, (which is the graph of 

v = ^/x, Fig. 2), that the subtangent is bisected at the vertex; deduce 

dy 1 

the value of the derivative. — == -. 

dx 2 |/jf 

9. The graph of the function y = ^{a % — x 2 ) is the circle 
x 2 -\-y 2 = a 2 . Deduce the value of the derivative from the property 
that the tangent is perpendicular to the radius. 

dy x 

dx \/{a 2 — x 2 )' 



CHAPTER II. 

Formula and Methods of Differentiation. 



IV. 
Character of the required For mules. 

42. The functions of an independent variable are expressed 
by means of a few simple functional symbols and their com- 
binations, either by algebraic operations, or in the form of a 
function of a function. It is the object of the present Chap- 
ter to establish the formulae for the differentials of the simple 
functions of one independent variable, and also the formulae 
by means of which we can apply these to the differentiation 
of the more complex functions formed by combining the ele- 
mentary forms. 

43. Of the latter class of formulae we have already found 
those for the sum and for the product of two variables, Arts. 
29 and 31. From the latter we shall see, in the next section, 
that all the formulae for algebraic functions may be derived. 
We shall, however, first give a method of deducing the de- 
rivative of the square by means of a functional equation, and 
shall derive from it another proof of the formula for the product. 

Differentiation of the Square. 

44. Let/(#) = x 2 ; it is required to find f{%). Assume 
another value of the independent variable connected with x 
by the relation 

z =mx y (1) 



§ IV.] DIFFERENTIATION OF THE SQUARE. 33 

where m is a constant. We propose to find a relation between 
the corresponding values of the derivative. 

Since the members of equation (i) remain equal while each 
is variable, their rates and consequently their differentials are 
equal. In other words, we can differentiate the equatio?i. 
Thus, by Art. 30, 

dz=- mdx (2) 

In like manner, from the relation between the functions, 
which is 

f=™ % J (3) 

we have, by differentiation, 

d(z 2 ) = ni'dix 1 ) (4) 

Dividing equation (4) by equation (2), we have 

dz dx 

a relation between the derivatives f{z) and/'(#) ; that is, be- 
tween the values of the derivative of the function " square " 
which correspond to two values of the independent variable 
connected by equation (1). These values are thus not inde- 
pendent of one another in equation (5); but, if we eliminate m 
by means of equation (1), thus obtaining 

1 A{f) = 1 d(x>) ^ ^ 

z dz x dx ' ^ ' 

or 

we have a relation in which x and z are entirely independent, 
because we obtain the same result no matter what the value 



34 FORMULA OF DIFFERENTIA TION. [Art. 44. 



of m. In this equation the variables are already separated, 
as in Art. 16; accordingly, the members are of the same 
form and constitute an expression which does not vary with 
x. Hence, denoting its constant value by c y we have 

~f^) = c; . (7) 

therefore 

/'(*) = -|r= cx * ( 8 > 

or 

d{x 2 ) = cxdx (9) 

Applying this to the identity 

(x -f- h) 2 = x 2 -f- 2 hx -f- h 2 t 

where h is a constant, we have 

c(x -\- h)dx = cxdx -j- 2hdx ; 

whence c = 2,f and equation (9) becomes 

^(x 2 ) = 2xdx (10) 

* Referring to the graph of this function. Fig. 1, p. 4, it will be seen that the 
meaning of the process is that, if two points move on the curve in such a manner 
that their abscissse have the constant ratio m, the rates of the abscissa? have the 
ratio w, and those of the ordinates the ratio m 2 . Hence the slopes or values of 
tan (p, see equation (5), have the ratio m, that is, the same ratio as the abscissae. 
That is to say, the slope is proportional to the abscissa. 

\ Had we commenced with the relation z = x -J- h we should have obtained 
d{z*) _ d{x>) _ 

~ m ^ m Z, At -.1 • ]ZiJC — W« 

dz dx 

in which the constant is easily shown to be zero. 



g IV.] THE DIFFERENTIAL OF THE PRODUCT. 35 

The Differential of the Product. 

4-5. If we apply the formula just derived for the square 
to the identity 

{x -\- y) 2 = x 2 -f- 2xy -f- y 2 > 

regarding the differential of the product xy to be as yet un- 
known, we find 

2(x -\- y) (dx -f- dy) = 2xdx -\~ 2d(xy) -j- 2ydy, 

which reduces to 

d(xy) — xdy -j- ydx. 

This formula is readily extended to products consisting 
of any number of factors. Thus, let x x x 2 x z . . . . Xp denote 
the product of p variable factors, then 

(1/ ( X-, XnXo • . . . Xj,J XaXn • • • Xj/U/X-, X i (J/ I X nX q • • • Xjf) 

XnXo • • • Xtf-vX] ~~ I X-,Xq > . • X fiU/Xn X-, Xn(J/K Xn ' . . Xj,) 

:== XyOC^ . . . XaLX-^ ~\~ X^X^ ... XpCLX^ • • . ~\~ X^ . • • X j,_,CLXj ) . \*5) 

Hence the differential of the product of several variables is the 
sum of the products of the differentials of the factors each 
multiplied by all the other factors. In other words, it is the 
sum of the differentials obtained by supposing each factor in 
turn to be the only variable one. 

Differentiation of an Inverse Function. 

46. When the derivative of a function is known, the 
derivative of the function inverse to it is readily deduced. 
Thus, in the case of the square root, which is inverse to 
the square, let 

y = j/x, whence x = y 2 . 



36 FORMULA OF DIFFERENTIA TION. [Art. 46. 

Taking the derivative of this function of y, we have 

dx dy 1 

-7- = 2v, whence -7- = — . 
dy dx 2y 

Finally, expressing this inverse derivative in terms of x, 

dy 1 dx 

or dy - 



dx 2\/x' 2\/x' 

that is, 

dx 
dJx = . 

r 2^X 

Differentiation of a Function of a Function, 

47. Since the ratio of differentials is independent of their 
actual values, % in the formula just obtained may be replaced 
by a variable which is not independent, and may not have a 
constant rate. Accordingly, the formula, when expressed as a 
rule, becomes : The differential of the square root of any varia- 
ble is the result of dividing the differential of the variable by 
twice the given square root. For example, if a 2 -\-x 2 is the 
variable which takes the place of x in the formula, we have 

d(a 2 + ^ 2 ) 2xdx x 

dV(a + &) = 2V (tf + a 2) = : 2 i /(a 2 +x 2 ) _ vitf+x*) 

4-8. So in general, for any " function of a function," or 
expression of the form 

y= [/(*)]» 

if the rules for the differentiation of the functions and f are 
known, we can differentiate y, that is, express dy in terms of 
x and dx. For, if z =f(x), we have y = 0(z), whence 

dy = <p'(z)dz = <p'(z)f'(x)dx, 

in which the functions 0' andf are supposed known. 



§ IV.] THE RATES OF GEOMETRICAL VARIABLES. S7 

It is for this reason that, as mentioned in Art. 43, it is 
only necessary to prepare formulae for the differentiation of 
the simple functions. 

The Rates of Geometrical Variables. 

4-9. Geometrical magnitudes dependent upon the posi- 
tion of a point become variables having definite rates, when 
the velocity and direction of the point are given. We have 
already employed the velocity of the moving point to repre- 
sent the rate of its distance from a fixed point in the line of 
its motion supposed straight. Such a distance, when it occurs 
in a problem, is thus marked out as the most convenient inde- 
pendent variable x, in terms of which to express any other 
variable y of which the rate is required. For, the rate of x 
will be known, and the derivative with respect to x is the 
relative rate of y. 

50. As an illustration, suppose a man to be walking on a 
straight path BC at the rate of 5 
feet per second: required the rate 
of change in his distance AP from a 
point A at the perpendicular dis- 
tanced^ = 120 feet from the path, 
at the instant when he is passing; the B 

Fto 8 

point C, 50 feet from the foot of 

the perpendicular. Denote the variable distance of the man 

dx 
from B by x, so that -=- may denote the known velocity of P, 

and denote the constant AB by a. Then, by geometry, the 
variable distance of which the rate is required is 

y= |/(a 2 -(- x 2 ). 



A 


\ 


a 


i \ 




x \ \ 



3§ FORMULA OF DIFFERENTIATION. [Art. 50. 

Hence its rate is 

dy _d |/(a 2 -f- x 2 ) _ x dx 

~di~ dt ~ \/(d* + x 2 ) ' ~dt' 

Substituting herein the special value x = 5, together with 

dx dy 

a = 120 and -7- = 5, we find -7- = iff, which is therefore the 

rate at which the distance AP is increasing at the instant 
P passes C. 

51. We might, in the solution of this problem, have ex- 
pressed y directly in terms of /. For this purpose, assume the 
instant when P passes B as the origin of time. Then the 
value of BP at the end of the time / is 5/, and that of AP is 

y = |/(I20 2 + 25/*) = 5 |/(2 4 2 + P), 
whence 

dy _ 5/ 

~dt~ V(576 + *)' 

The man is at C when t= 10; therefore the rate re- 
quired is 

dyn 
dt_ 
as before. 

Again, we have 



26 ~ Iyir ' 



Sd = °' and * 



5. 



Of these results, the first shows that when P is at B it is 
neither approaching nor receding from A ; and the second 
shows that the rate of receding from A has for its limiting 
value the actual velocity of P, or rate at which it recedes 
from B. 



§ IV.] EXAMPLES. 39 

Examples IV. 

i. Differentiate {2.x -f- 3)*, and find the numerical value of its rate, 
when x has the value 8, and is decreasing at the rate of 2 units per 
second. 

The differential required is denoted by d [(2.2: -j- 3) 2 ]? an d the rate by 

4(2X+ 3 ) 2 ] . . dx 

_L__ h the given rate y = - ». _ 1$2 units per ^^ 

2. Find the numerical value of the rate of (x 2 — 2x) 2 , when x = 3 
and is increasing at the rate of ^ of one unit per second. 

Differentiate the given expression before substituting . 

12 units per second. 

3. Find the numerical value of the rate of \/{y 2 + x2 )> when y = 7 
and x = — 7, if jyis increasing at the rate of 12 units per second, and 
x at the rate of 4 units per second. 

4 |/2 units per second. 

4. If f(x) = x — ^{x 2 — a 2 ), find/^), and show that f(x) is a 

decreasing function. _,, . x 

f(x) = 1 - 



\/{x 2 — a 2 ) 

5. Differentiate the identity (\/x -\- j^a) 2 = x -{- a -{- 2 \/ax, and 
show that the result is an identity. 

^.„ . / (x 2 — 2ax 

6. Differentiate!/ — = T 

" \a 4 — 2ao 

The constant factor — — -z — should be separated from the vari- 

^{a 1 — 2ao) 



able factor before differentiation. 

1 x — a 



/\/{a 2 — 2ab) j^{x 2 — 2ax) 



dx. 



x 



7. ii/{pc) = (1 + *)*, f'{x) = (i + x%) i 

8. If f(x) = |/(a 3 + 2Px + ex*), 

f>l x \ - P + CX 

~~ V( aS + 2 ^ x + cx ') 
9. if f{x) =y[> + y (l + x>)l f{x) = ' ^/{.y 1 - 



40 FORMULAE OF DIFFERENTIATION. [Ex. IV. 

10. if Ax) = — * ,. , /'W - i + 



Rationalize the denominator before differentiating. 

x v^ dv 

ii. Given — 2 -\-~^ = i? express — in terms of x, and give the 



dy 

values of — 

dx 



a d y~ 

and — 
dx 



dy b x 



dx a ^/{a 2 — x 2 ) 

dy . 
1 2 . Given jr = \ax, express — in terms of x, also in terms of y, 



dy 
and give the values of — 



and — 
, dx 



dy /a 2a 

13. If the side of an equilateral triangle increases uniformly at the 
rate of 3 ft. per second, at what rate per second is the area increasing, 
when the side is 10 ft. ? I 5 \^3 s q- ft. 

14. A stone dropped into still water produces a series of continu- 
ally enlarging concentric circles ; it is required to find the rate per 
second at which the area of one of them is enlarging, when its diame- 
ter is 12 inches, supposing the wave to be then receding from the 
centre at the rate of 3 inches per second. 36 n sq. inches. 

15. If a circular disk of metal expands by heat so that the area A 
of each of its faces increases at the rate of 0.0 1 sq. in. per second, at 
what rate per second is its diameter increasing? 

r 

; TV in - 

100 /^{jtA) 

16. A man standing on the edge of a wharf is hauling in a rope 
attached to a boat at the rate of 4 ft. per second. The man's hands 
being 9 ft. above the point of attachment of the rope, how fast is the 
boat approaching the wharf when she is at a distance of 1 2 ft. from it ? 

5 ft. per second. 

17. A ladder 2 5 ft. long reclines against a wall ; a man begins to 
pull the lower extremity, which is 7 ft. distant from the bottom of the 
wall, along the ground at the rate of 2 ft. per second ; at what rate 
per second does the other extremity begin to descend along the face 
of the wall ? 7 inches. 

18. One end of a ball of thread is fastened to the top of a pole 35 



§ V.] THE DIFFERENTIAL OF THE RECIPROCAL. 4 1 

ft. high ; a man holding the ball 5 ft. above the ground moves uni- 
formly from the bottom at the rate of five miles per hour, allowing the 
thread to unwind as he advances. What is the man's distance from 
the pole when the thread is unwinding at the rate of one mile per 
hour? I |/6 ft. 

19. A vessel sailing due south at the uniform rate of 8 miles per 
hour is 20 miles north of a vessel sailing due east at the rate of 10 miles 
per hour. At what rate are they separating — (a) at the end of 1^ 
hours ? (/?) at the end of 2 -J- hours ? 

Express the distances in terms of the time, (a) 5^ T miles per hour. 

20. When are the two ships mentioned in the preceding example 
neither receding from nor approaching each other ? 

When / = J^ of an hour. 



V. 

The Differential of the Reciprocal. 

52. The differential of the reciprocal is readily obtained 
by means of the implicit form of this function. 
Denoting the function by y, we have 

1 
y = — , .*. xy= 1. 

J x J 

Differentiating the latter equation by the formula for the 
product, we obtain 

ydx -f- xdy = o, 
whence 

. ydx 

dy =- — ' 

substituting the value of y, 




42 FORMULA OF DIFFERENTIATION. [Art. 52. 

That is, the differential of the reciprocal of a variable is 
the negative of the result of dividing the differential by the 
square of the variable. 

This formula enables us to differentiate any fraction of 
which the numerator is constant and the denominator a varia- 
ble whose differential is known. Thus if 

a 2 - b 2 
y ~a 2 -oP 



we have 



dy = (a 2 - V) d^-j = (o» - V) J X _ * 



Differential of the Quotient of Two Variables. 

S3. Since a fraction of which both terms are variable 
is the product of its numerator and the reciprocal of its 
denominator, we can now express the differential of such a 
fraction in terms of those of its numerator and denominator. 
Thus 

dl — ) = dl x — )= — dx-\- xd 

\y/ \ y/ y 




_ dx xdy _ ydx - . 

~~ ~y f~ ~~ f " " " ." 

That is to say, the differential of a fraction is the result of 
taking the product of the denominator into the differential of 
the numerator minus that of the numerator into the differen- 
tial of the denominator , and dividing by the square of the 
denominator. 



§ V.] DIFFERENTIA TION OF X n . 43 

The signs of the terms in ydx — xdy can be recollected by- 
recalling the fact that a fraction is an increasing function of 
its numerator and a decreasing function of its denominator. 

As an illustration of the application of this formula, we 
have 

(2x — a\ 2(x % -\-b) — 2x(2x — a) b-\-ax—x 2 

d [^f^J = (^ + b f dx = 2 ^ + b y dx - 

Differentiation of x n . 

54. To obtain the differential of the power when the 
exponent is a positive integer, suppose each of the variables 
x v x v . . . Xp in formula (B), Art. 45, to be replaced by x. 
The first member contains p factors, and the second p terms ; 
the equation therefore reduces to 

d{x p ) = px^-Hx. . . . . . (1) 

p 
Next, when the exponent is a positive fraction, let n = — , 

where p and q are integers. Put 

t 
y = x n = x q , whence y q = x*. 

This last equation can be differentiated by equation (1), be- 
cause p and q are both integers. Thus 

qy g ~ 1 dy = px p ~^dx, 
whence 

dy = — • — -1 dx. 

q y 



44 FORMULAE OF DIFFERENTIA TION. [Art. 54.. 

Substituting the value of y, 

^5/) = —. zdx——x? ax. . . . (2) 

Finally, when the exponent is negative, we have 

1 
x~ m = — . 
x m 

Differentiating by the formula of Art. 52, we obtain 

d{x m ) 



d(x~ m ) = — 



x* m J 



and, since wis a positive integer or fraction, we have, by (1) 
or (2), 

mx m ~ x dx 
d( X -'") = - %2m = - mx-™~Hx. . . (3) 

Equations (1), (2) and (3) show that, for all values of n, 

d(x n ) = nx n ~ 1 dx. . . . . . (a)* 

55. Formula (a) includes those already found for the 
square, the square-root, Art. 46, and the reciprocal, Art. 52, 
which are the special cases corresponding to n =2, n = \ 
and n = — 1. The two last-mentioned formulae are, how- 
ever, particularly useful because applicable to a familiar form 
of notation different from that of fractional and negative 
exponents. 

* The formulae of this series are recapitulated on page 79. Together with 
formulas (A), (B) and (C), for the algebraic combinations of functions, they form 
the body of rules for calculating differentials which properly constitute the Differ- 
ential Calculus. 



§ V.] EXAMPLES. 45 

On the other hand, it is often useful to transform an 
expression, by the use of fractional and negative exponents, 
in order to employ the general formula (a) instead of a com- 
bination of the special ones. Thus 

Again, 

I 3 5 

—-, — - - 3 = d(a + x)~* = — ¥a -f x)~~*dx. 

_ 4/(0+ x) 6 J J i\ \ j 

The derivative of a function may be written at once in- 
stead of first writing the differential, since the former differs 
from the latter only in the omission of the factor dx, which 
must necessarily occur in every term. Thus, given 



x 



we derive 



= (I + X 2 )-? — ix(l + X 2 )~% . 2X = 



•n». 



dx (i+x 2 )* 



i. Differentiate 



Examples V 

a -j- bx -j- ex 2 



x 



Put the expression in the form \-b -\- ex. \c ) dx. 

x \ x 2 / 

Find the derivatives of the following functions : 

a 2 —b 2 dy 2X 



a i -x i dx v \a 2 -x 2 ) 2 ' 

3-J>= V( x * -<**)> 4- = — Trr — sv- 

dx 2 \/[x 3 — a 3 ) 



46 FORMULA OF DIFFERENTIA TION. [Ex. V. 

2X i dy Ax 3 (2a 2 — x' 4 ) 

4. y = . — = -. 

a 2 — x 2 dx (a 2 — x 2 ) 2 

dy 

5. y — (1 + 2X 2 ){j -\- 4^8). — = 4^(1 + 3 x + io* 3 ). 

6. j/ = (# 3 + j; 3 )(^ 2 -(-3^ 2 ) -7- =3(5-^ 3 + &* 2 + 2a 3 )jt*. 

7 . y = (i -\-x)\i + x 2 ) 2 . 

£= 4(i+^) 3 (i+^)(i+^+^ 2 ). 

8. j/ = (1 + jv w ) m + (1 + x n ) m . 

— = w«[(i 4-jv w )»- 1 ^'«- 1 -f (1 -}-x n ) m - 1 x n - 1 ]. 



x 2 — 2a 2 




dy a 2 

— = 1 -4- 

dx (x — a) 2 ' 


9. y — 

x —a 


a — x 




dy a -J- x 


10. y — . 


ax 2x^ 


a/(x 2 — a 2 ) 




dy a 2 


1 1 . y — 

X 


dx x 2 ^/(x 2 — a 2 )' 


ab 




ab 2X 2 — a 2 


x " ^ " ex \/{x 2 — a 2 )' 


c x 2 {x 2 — a 2 )^' 


1 . 1 







**•*& _ 3 3 

— = J[(I -jp) -W- (!+*)-!]. 

/ , \ , , \ 4y l — ?> x 

14. y =(1 4- x) \/(l — x). ~= — -. 

<t s \ -r j v \ j dx 2 tf(i — x) 

15. y = (a -f- ^) 3 ( 3— xy% 2 . 

dy 

— — ^(0 -|- x) 2 (b — x) z [2ab -\- ($b — 6a)x — gx 2 ~\ . 

x n -f 1 ^ 2nx n ~ 1 

x n — 1 <£t; ~ (#* — i) 2 ' 



16. >> = 



17. j/ = (3^ -j- 2ax)Kb — #.#). -y- = — 5# 2 .*: 4/(3^ + 2ax). 



v.] 



EXAMPLES. 



47 



18. y 



19. y 



" t — v. 



I — X 

X 



dy 



dx (1 — x) |/(i — x 2 )' 



y (a 2 -\- x 2 ) — x* 
Rationalize the denominator. 



dy 



V nl 



a* -f- 2X 



20. y = 

21. y = 

22. y = 



x 



dx a 2 \__ \/(a 2 -\~ x 2 ) 
dy 



z^+ 2X 



^{a 2 - x 2 )' 

dx 
j^{2ax — x 2 )' 

a 2 -b 2 



{jzax — x 2 )? 
See Art. 55. 

x \/(a -\-x) 



dx ( a i _ ^2)f 

dy abx 

dx (?.ax — x 2 )i' 



d y / 2 H\ - x — a 

ax \2ax — x 2 p 



23- y = 

24. y = 

25- y- 

26. y = 

27. y — 



\/a 



x 



^a — ^(a — x)' 



x" 



1/(1 — x*y 

x s 

3* 

(1 — x 2 Y 

x n 
(1 + x) n ' 
I 
{a + x) m {b + x) n ' 

2X 2 — I 

28. y = -. ^. 

^ X\/(l-\-X 2 ) 



dy ix 2 2 — x 3 

^;~ 2 (1— ^«)t 

^ 3jf 2 

"-£ ^1 — x 2 )% 



dy 



nx 



« — 1 



29.^ 



-a: 2 ) 



.# 






3°- y 



-u 



1 — x 2 



(1 + * 2 ) 3, 



a& — (1 + x) n + v 
dy na -|- mb -f- (m -f- #).r 

^V 1 -f 4X 2 

dx x 2 (i+x z ) ie 

|/(l+^)+ j/(i-x 2 ) 

X 2 \/{l — x i ) 
dy 2X 3 — 4.x 

dx" ( x _ x 2)i^ ^-jk 2 )!' 



48 



FORMULA OF DIFFERENTIATION. [Ex. V. 



31. y = x(d 2 4- x 2 ) \/(a 2 — x 2 ). 



dy a* -\- a 2 x 2 — 4.x* 
dx ~ \/{a l — x 2 ) 



x' 



32. y- 



33- y 



34. y — 



(i + x>) 



2\n' 



I — X 



4/(1 4- x 2 )' 



x° 



dy 
dx 

dy 
dx 



2UX 



,2« -1 



(1 -\- x 2 ) n + v 

1 -\- X 
(i-fAT 2 )!' 



jv 4- |/(i + x 2 )' 
See Example 19. 






36. y = 



V(l+^ 2 )- 


- VC 1 


- X 2 )' 

dy 
dx 


Jf 




dy 



\/{x 2 -\- a 2 ) — a 



dx 



x 



X' 



4J*: 4 4- 3-x -2 



1 + 



4JV 3 . 



1 + 



Y(i - ^)_ 



V^+OJ 



37. Two locomotives are moving along two straight lines of rail- 
way which intersect at an angle of 6o°; one is approaching the inter- 
section at the rate of 25 miles an hour, and the other is receding 
from it at the rate of 30 miles an hour ; find the rate per hour at 
which they are separating from each other when each is 10 miles 
from the intersection. 2 \ miles. 

38. A street-crossing is 10 ft. from a street-lamp situated directly 
above the curbstone, which is 60 ft. from the vertical walls of the 
opposite buildings. If a man is walking across to the opposite side 
of the street at the rate of 4 miles an hour, at what rate per hour does 
his shadow move upon the walls : (a) when he is 5 ft. from the curb- 
stone? (/?) when he is 20 ft. from the curbstone? 

(a) 96 miles; (/?) 6 miles. 

39. Assuming the volume of a tree to be proportional to the 
cube of its diameter, and that the latter increases uniformly, find 
the ratio of the rate of its volume when the diameter is 6 inches to 
the rate when the diameter is 3 ft. -^. 



§ VI.] THE LOGARITHMIC FUNCTION. 49 

40. If an ingot of silver in the form of a parallelopiped expands 
roVo P art °f eacn °f i ts linear dimensions for each degree of tempera- 
ture, at what rate per degree of temperature is its volume increasing 
when the sides are respectively 2, 3 and 6 inches? 

If x denote a side, dx may be assumed to denote the rate per degree 
of temperature. -^-^ of a cubic inch 



VI. 

Differentiation of the Logarithmic Function. 

56. The logarithm of x to the base b is the value of y in 
the equation x= b y and is denoted by \og b x. In finding its 
derivative regarded as a function of x, we shall employ the 
method illustrated in Art. 44 in the case of the square. 

Assuming another value z of the independent variable 
connected with x by the relation 

z = mx y (1) 

where m is a constant, the fundamental property of the func- 
tion is expressed by 

log z = log m -\- log x, (2) 

the symbol for the base being omitted for the present. Dif- 
ferentiation of equations (1) and (2) gives 

dz = mdx, d(\og z) — d(log x) ; 

whence, by division, we have 

d(\ogz) ^ 1 d(logx) ^ 

dz m dx ' '" ^ 



50 FORMULA OF DIFFERENTIA TION. [Art. 56. 

for the relation between the values of the derivative corre- 
sponding to two values of the independent variable connected 
together by equation (1). 

Now, eliminating m by means of equation (1), we obtain 

J(\ogz) _ d(\og x) 

Z dz ~ X ~~dx~~> ..... (4) 

in which the variables z and x are entirely independent, be- 
cause we arrive at the same result, no matter what the value 
of m. In other words, we have shown that, if f(x) = log#, 



zf\z) = xf{x). 



-X- 



It follows, as in Art. 16, that each member of this equa- 
tion has a value independent of x. Hence we write 

X -dx—= B > (» 

in which we have denoted the " constant" by B, because, 
while it is independent of x, it is obviously not independent 
of the base b of the system of logarithms. 

57. We have next to derive a functional relation between 
B and b. For this purpose, let a be another value of the base. 
From equation (5) 

Bdx 
d(\og b x) = — — , (6) 



x 



and therefore also 



d(\og a x) = ——, (7) 



x 



* In the graph of the function _y = log x, this equation signifies that the value 
of tan (p, or the gradient of the curve, is inversely proportional to the abscissa. 
See Fig. 9, p. 53. 



§ VI. J THE LOGARITHMIC FUNCTION. 5 1 

in which A is the same function of a that B is of b. 
Since by the definition of the logarithm 

x = b log * x , 

we have, by taking logarithms to the base a, 

log* oo = log, b . log 5 x) (8) 

whence, differentiating by equations (6) and (7), 

Adx Bdx 

— = l °Sa b — ~, 

or 

* A = B log, b. 

Hence, by the properties of logarithms, A = log, b B , and 

b B =a A . 

The independent quantities* b and a are here separated, so that, 
as in Art. 16, the common value of the members is independent 
of a or b. In other words, it is an absolute constant, and 
denoting it by e we have 

b B = e (9) 

Now, taking e as the base of a system of logarithms, we 
have 

B \og e b = 1 ; whence B = 



log, &' 



52 FORMULA OF DIFFERENTIATION. [Art. 57. 

Finally, substituting in equation (6), 

_d 



dx 
d{\og 5 x) = /v . ] h (b) 



Napierian Logarithms, 

58. The constant e is an incommensurable quantity second 
only in importance to the constant n. It is known as the 
Napierian base, and the corresponding system of logarithms 
as the natural or Napierian system. The method of comput- 
ing its value to any required degree of accuracy will be found 
in a subsequent chapter. 

Putting e in the place of b in the general formula (b), we 
have the special case 

dx 
d{\og e x) = — . (b f ) 

X 

Thus the natural logarithm is that which has the simplest 
derivative.* On this account the logarithms employed in 
analytical investigations are almost exclusively Napierian. 
Whenever it is necessary, for the purpose of obtaining numeri- 
cal results, these logarithms may be expressed in terms of the 
common tabular logarithms by means of the formula 

lo gio x ~ lo gio e lo ^ *» 

which is derived from equation (8), Art. 57, by writing 10 for 
a and e for b. The value of the constant log 10 e will be com- 
puted in a subsequent chapter. 

* The ground upon which Napier, the inventor of logarithms, chose the natural 
base is equivalent to the assumption that x and log x shall, in starting from their 
initial values I and o, begin to vary at the same rate. 



VI.] 



NAPIERIAN LOGARITHMS. 



53 



Hereafter, whenever the symbol log is employed without 
the subscript, log, is to be understood. 

59. The graph of the function log x, or curve whose 
rectangular equation is 



y = log, x, 

is called the logarithmic curve. 

The shape of this curve is indi- 
cated in Fig. 9. It passes through -*, 
the point A whose coordinates are 
(1, o), since 

log 1 = o. 
Since we have, from formula (&'), 



Fig. 9. 



(1) 




tan = 



dy 

dx 



x 



(2) 



the value of tan at the point A is unity, and therefore AC, 
the tangent line at this point, cuts the axis of x at an angle of 
45°, as in the diagram. It follows from equation (2) that 



when 


x > 1, 


tan < 1, 


and when 


x < 1, 


tan > 1 ; 



the curve, therefore, lies below the tangent AC, as shown in 
Fig. 9. 

The point (e, 1) is a point of the curve; let B, Fig. 9, be 
this point, then OR will represent the Napierian base, and 
BR = 1 = OA. Produce BR to meet the tangent in C; then, 
because the tangent lies above the curve, RC > 1. But since 



54 FORMULA OF DIFFERENTIATION. [Art. 59. 

RAC = 45°> AR = RC; hence AR > 1 and OR > 2; that is, 
the Napierian base e is somewhat greater than 2. 



Logarithmic Differentiation. 

dx 
60. The expression — is often called the logarithmic dif- 

X 

ferential of x. When the value of x is positive, it is by the 
preceding articles the differential of the Napierian logarithm of 
x. But, when x is negative, so that the logarithm is imaginary, 
the logarithmic differential is still real, and is in fact then the 
differential of the logarithm of the numerical value of x taken 
positively : for 



d[log(—x)] = 



d{ — x) dx 

X X 



Hence we may define the logarithmic differential as the dif- 
ferential of the logarithm of the numerical value of x regard- 
less of algebraic sign. 

The complete graph of the function corresponding to the 
logarithmic differential would consist of the curve of Art. 59, 
together with the dotted branch represented in Fig. 9. 

61. By the process of logarithmic differentiation we may 
derive independent demonstrations of the formulae already 
found. Thus, by differentiating the equation 

log (xy) = log x + log y, 



* The logarithm of a negative quantity is imaginary ; but, by the properties of 
logarithms, we must have log (— jr) = log x -f log ( — 1). The term log (— 1) con- 
stitutes the imaginary part; but, since it is a constant, the differential of log(— x) 
is the same as that of log x. 



§ VI.] LOGARITHMIC DIFFERENTIATION. 55 

we have 

d(xy) dx dy 

— 1 ; 

xy x y 

whence 

d(xy) = ydx -\- xdy. 

In like manner, from 



we derive 



whence 



lo g(- ) = logx — logy 



y I x\ _ dx dy 



x \y / x y 
Ix \ __ ydx — xdy 





~\y 1 ~ f 


Again, from 






log x n = n log x 


we have 






d(x n ) dx 
x n x 



whence 

d(x n ) = nx n ~ 1 dx. 

62. The method of logarithmic differentiation may fre- 
quently be used with advantage in finding the derivatives of 
complicated algebraic expressions. For example, let us take 

|/(2S)(I - tt 8 )* 

» = ; rr - » (0 

[X — 2) 3 



$6 FORMULAE OF DIFFERENTIA TION. [Art. 62. 

whence we derive 

log u — -J log (2x) -f- 1 log (i — x 2 ) — f log (# — 2). (2) 

Differentiating, 

du 1 $x 2 

udx ~~ 2x 2(1 — x 2 ) 3(3; — 2) ■' ^ 3 ' 

adding and reducing, 

du _ — 8^ 3 -\- 24^ — # — 6 

wd# 6(1 — X*)(X — 2)x ' 

therefore 

dw _ — 8x 3 -f- 24^ — x — 6 
^ 3(2x^(1 — x*)*{x — 2)* ' 

Differentiation of Exponential Functions. 

63. An exponential function is an expression in which the 
exponent is variable. In the simple exponential function 

y=a* . ' (1) 

it is necessary to assume a to be positive ; otherwise y is not 
a continuous variable. This is the same thing as saying that 
in the inverse of this function, x = log a ^y, we cannot have a 
negative base for a system of logarithms. 

The differential may be obtained from the inverse function, 
as in Art. 46, or as follows, which amounts to the same thing. 
Taking Napierian logarithms of both members of equation 
(1), we have 

log y = x log a ; 



VI.] 



EXPONENTIAL FUNCTIONS. 



57 



differentiating by formula (b f ), 

dy 



hence 



or 



= log a . dx ; 

dy = log a . y dx, 
d{a x ) = log a . a x dx. 



(c) 



case 



64. Putting e for a in this formula we have the special 

d(e x ) = e x dx ■ . . (c') 



The exponential of this form is the most common in analy- 
sis. It is often denoted by the functional symbol exp, espe- 
cially when the exponent is a complicated expression, for 
example, exp[x \/(x 2 — i)]. 

Formula (c') shows that e* is the function which is its own 
derivative. Its graph, Fig. io, is called 
the exponential curve, and is the same 
as the logarithmic curve in another posi- 
tion. If from any point P of a curve 
• the tangent PT and ordinate PR be 
drawn, the subtangent TR = y cot <p. 
Now in this curve, tan = e x =y, we 
therefore have y cot <f> = ■ I. Hence the 
exponential curve is the curve in which the subtangent is con- 
stant. In the diagram, this constant value is equal to OB, 
which represent unity. 

65. When both the exponent and the quantity affected by 
it are variable, the method of logarithmic differentiation may 
be employed. Thus, if the given function be 




Fig. io. 



Z = (nx) x \ 



5 8 FORM ULM OF D IFFERENTIA TIOJV. [Art. 65 

we shall have 

log z = x 2 log (nx) ; 

differentiating, 

dz .dx 

— =x z — + 2x log (nx)dx. 

Z X ' & v ' 

hence 

d[(nx) x *] = (w^) x2 x[i -(- 2 log (ft#)]J#. 



Examples VI. 

dy~ 
1. Given the function^ = log^jc; show that — 



toga* , 

1 , and 

e 

hence prove that the tangent to the corresponding curve, at the point 

whose abscissa is e, passes through the origin. 

dy 
2. y — x n log x. ~ — x n ~\\ -f n log x). 



3. y = log (log *). 

4- jy = log [log(a + &* M )] . 

5. j/ = 4/^ — log (4/* + 1) 

6. y = log 



^ 



^ or log x 
dy _ nbx n ~ l 

dx~ (a -f- &r M ) log (a -f- far*)" 
^ 1 

tffo; 2(|/jf-f- 1) 

|Az — \/x' dx {a — x) \/x' 

Put in the form log (\/a -f- 4/x) — log ( ^a — \/x). 

1 ._y = lo g [W X -a) + V ( x -i)]. ! = __L_^. 
8.., = tog [*+•<*■**)]. | = _^_ r) . 

J? <^/ I 



9. j/ = log 

10. y = log 



|/(i -j- x 2 ) dx x(i -J- x 2 ) 

|/(l -f- x) -f- |/(i — •#) ^ _ * 

|/(i -j- jt) — 4/(1 — jf)" dx x |/(i — jv 2 )' 



1 1. ^ = log [*+ 4/(a 2 -^)] . ^ = 



dx ^(a 2 — x 2 )[x -j- \/(d 2 — x 2 )~\ 



§ VI.] EXAMPLES. 59 

. x dy i t 

12. y= log -. T~=-+ 



. . aizx — d) dy x % 4- a 2 

ia. y =. log ( x — a) — -f r-5-. — = ! . 

4 ^ 5 V } (x-af dx (x - af 

1 5 . y = a* . -— = 2 log a.a x x. 

D dx & 

-^- ^/ i J__ 

10. Vr=^l + *. -=— = £ 1+ *. 

</y 

17. j/ = 6T*(l — X 3 ). — - = g*(i — 3^ — ^ 3 ). 

18. _y = (.r — 3)* * -f- 4^e*. — = (2jr — 5)<? 2 * 4- 4(jr -f- i)e x . 

_ e x — e~ x dy 4 

20. _y = K- ^= log « • log b . b a . a*. 

*-« ^ « «-i i 

21. v = a x . — — = «tz^ . x n 1 . log <2. 

dx 
x dy e*(i — x) — 1 



22. y = 



e* — 1 dx (e x — i) 2 

dy e* — e~ 



.23. jk = log (** + *"*). 



^ £* -j- £ ■* 



24. jy = tf log *. -/=-logtf. <z Io s*. 



25. y = log 



e* dy 



I+C* <£*r ~ I -|- ** 



26.y=x*. — — jr* (1 -f-log.r). 

28. y =£**• 2£ =0** . ^*(i _f- log *). 



60 FORMULAE OF DIFFER ENTIATION. [Ex. VI. 

dy \ i -log* 

— = x 3 — * 

dx x 

dx 

^L =x e* c* }+ xlosx . 

dx " x 

— = (log a) 2 xa x > 
dx 

dx 



. , , dy (x- i)'(7 -y 2 + 3°-^-9 7) 

See Art. 62. -=- = -7- ro * 

"^ 1 2 (o: — 2 )*(.*: — 3) 3 

_ +/[ax(x — 30)] <z> -jA* (~r 2 — 8«a- -f 12 a 2 ) 

35 * ^ |/(^ — 4«) dx" 2 [ x (x — 3a)]i(x — 4^)*' 

__ (■*+!)*(■* + 3)* # = **(.*+ 3)* 

3 *- r " (^+2) 4 <&? (a:-|-2) 5 (A-+ i)i* 



1 




29. y = x x . 




e x 

30. y = e . 




31. y = xe*. 




32. jy = a x (x log # — 1). 




33. y — 2e Vx (xi— $x -\- 6x%- 


• 6). 


(x- i)t 




34- y — , ,3, .1 





37- .7 = 



{% — 2) 9 ^ (JT — 2) 8 (X 2 — 7JT -f- i) 



(x — i)?(x — 3)~ff~ ^ (^ — i) 3 (x — 3) 



1 3 

2 



1 *$ 

(^ 2 — 2Jtr + 2)1 (x? + 1)* 
38. y =■ . 

(•* + J )* 

<7jy _ (8-^ S — 2TJf 2 -\- 26X — i$){x 2 +1)^ 
" X 2 (x 2 — 2 Jf -j- 2 y*{x — j— I )a 

_ exp[^r \/{x 2 — 1)] </y _ 2 -|/(^: 2 — 1) exp[jr |/(j; 2 — 1)] 

39* jk: -f- |/(~r 2 — 1) ^ x+ |/(jt 2 — 1) 

40. G 1 ven^ = ^ + v(i _^ ) j, prove — =___^. 
4 1 - Given * = 4 /( I -^) ( I + 4 /(i-^) )' Pr0VC ^ 

fifo / X \ n I -\- 7t ^/(l — X 2 ) 

dx \i + 4/(1 — -* 2 )/ (1 _ x ?)% 

Put ( ; 5r ) = y. and use the result obtained in Ex. 40. 

\i + V(i —x 2 )) 



VII.] TRIGONOMETRIC OR CIRCULAR FUNCTIONS. 



61 



VII. 

Trigonometric or Circular Functions. 

66. As stated in Art. 7, when the trigonometric func- 
tions, sin 0, cos etc., are regarded as undergoing continuous 
variation, the independent variable 6 is taken to be the arcual 
measure of the angle, that is, the ratio which the arc sub- 
tending the angle at the centre of a circle bears to the radius. 

Let be the centre of a circle of radius a, referred to rect- 
angular diameters as coordinate axes, and let A OP, Fig. 11, 
be the angle 6, the radius OA being fixed in the axis of x. 
Denote the arc AP by s, then 6 — s/a, and as increases P 
moves along the circumference, completing the circuit (and 
the radius OP completing a revolution), when '= 27t. Since 
s may increase indefinitely with 
repeated revolutions of P, the 
trigonometric functions defined 
by the' ratios of the sides of the 
right triangle OPR are continuous 
functions for all values of 6. 
They are called periodic functions, 
because their values repeat them- 
selves while 6 passes through suc- 
cessive ranges of values, each of 
extent 2n\ hence also 2n is called Fig. ii. 

the period of the functions. Compare the graphs in Figs. 12 
and 13, pp. 63 and 64. 

Since both the functions and the independent variable are 
defined by means of lines connected with a circle, they are 
often called circular functions. We shall derive their rates of 
variation from their geometrical definitions. 




62 FORMULAE OF DIFFERENTIA TION. [Art. 6?* 

Differentiation of the Sine and the Cosine. 
67. We have then, by definition, in Fig. n 

= -, sin0=A cosfl = -. . . . (i) 

a a a 

Let P.P, measured along the tangent line in the direction in 
which P moves when 6 is increasing, represent ds; then 
drawing PB and BP' parallel to the axes, we complete, as in 
Fig. 7, the differential triangle for the motion of P. From 
their directions in Fig. 1 1 it appears that BP' represents dy, 
which is positive because y is increasing, and BP represents 
— dx because x is decreasing, so that dx is negative. 
Differentiating equations (i), we have 

d8=~, d (sin 6) = ^, d(cos 0) = - ; 

a a a 



whence the derivatives of sin 6 and cos 6 are 

d{sin 6) _ dy d(cos 6) _ dx 

dd ds dd ds' 



(2) 



To express these in terms of 6 y we note that the differential 
triangle and OPR are similar because their sides are mutually 
perpendicular, the tangent to a circle being perpendicular to 
the radius drawn to the point of contact.* 

* This can be shown independently of geometry by differentiating the equation 
of the circle, which is 

x 2 -\-y 2 — a 2 -, 
for this gives 

xdx-\-ydy = o, 
whence 

dy x 
— dx y * 



5 VII.] 



THE SINE AND THE COSINE. 



63 



Therefore the angle BP P is equal to #, and 

sin 6. 



dy dx 

■y-= COS 6, -J- = 

ds ds 



Substituting in equation (2), we have 



and 



d(sin 6) = cos 6 dd . 
d(cos 6) = — sin Odd. 



id) 



68. The arcual measure is the length of the arc in the 
circle whose radius is unity. In the graph of the function 
sin x, this is measured along the axis of x. The curve 
y = sin x is the full 
line in Fig. 12, which is 
a continuous curve of 
unlimited extent, con- 
sisting of repeated simi- 
lar branches. Since 
d(sin x) - 



dx 



Y 












,-— 


>< V 








,• 





/ \ 




\F 


/ 


'' 
















\ 






y 








"■■ 









#T> X 



= cos 0=1, 



Fig. 12. 



this curve makes an angle of 45 ° with the axis of x at the 
origin. The graph of cos x is the dotted line in the diagram, 
which is the same curve moved a distance equal \n to the 
left, since cos x = sin (x + \7t). 



Differentiation of the Tangent and the Cotangent. 

69. The differential of tan d is found by applying formula 
(C), p. 42, to the equation 

sin d 



tan = 



cos d ' 



6 4 



FORMULAE OF D IFFERENTIATION. [Art. 69, 



thus, using formulae (d) and (e), 



cos 2 d + sin 2 JQ dd 

</(tan 6>) = — -^r- -dd = 



cos 2 d 



cos 2 # 



2/) » 



or 



d(tan 0) = 



dd 

cos 2 d 



c ~~2 



sec 2 d dd (/) 



The differential of cot can be obtained in a similar man- 
ner, or by applying this formula to the equation 



which gives 



cot =.tan(i*r — 6) ; 



J(cot 0) = — 



dd 



sirfd 



= — cosec 2 dd. . . (g-) 



70 B The function tan x is discontinuous for any range of 
values including an odd multiple of J7r, because the value of 

the function is infinite when x = ± ^7T, 
± f 7T) etc. Accordingly, the graph or 
curve y = tan x, given in Fig. 13, con- 
sists of an unlimited number of de- 
tached branches, each corresponding to 
a range of values of x of extent n. The 
period of this function is therefore n. 
Each branch cuts the axis of x at 
Fig. 13. an angle of 45 °, and at these points the 

curve has, by formula (/), its smallest gradient. 

Differentiation of the Secant and the Cosecant. 

71. The differential of sec d is found by applying the rule 
for the reciprocal to the equation 

1 




sec 6 = 



cos d 1 



§ VII.] THE SECANT AND THE COSECANT. 



65 



thus 



„ x sin Odd 
d(sec 6) = t^t = sec tan do. 

v COS 2 /7 



. (A) 



The differential of cosec is found by applying this 
formula to the equation 



which gives 



cosec 6 = sec (J-tt — #), 
J(cosec 6) = — cot 2 6 dd. 



(0 



The graph of the function 
sec x is given in Fig. 14. It con- 
sists of discontinuous branches 
'alternately above and below the 
axis of x. The period of the 
function, 27r, corresponds to two 
branches. 

72. To these formulae (d) to 
(z) for the six trigonometric func- 
tions may be added that for the 
versed-sine, which in Fig. 11 is 
the ratio of AR to the radius OP. 





Fig. 14. 



This function is therefore defined by the equation 



vers 6=1 — cos d> 



whence, by formula (e), 



J(vers 6) = sin Odd (j) 



66 FORMULA OF DIFFERENTIA TION. [Art. 73. 

The Logarithmic Trigonometric Functions. 

73. Combining the formulae found above with that for the 
logarithm, we readily derive the following for the logarithms 
of the circular functions: 

d(log sin 6) = — d(log cosec 6) =cot 6dd; 

d(log cos 6) = — d(\og sec 6) = — tan 0^0; 

d(\og tan 0) = — d(\og cot 0) = (tan + cot d)dd. 

Examples VII. 

1. The value of d{s\n 6) being given, derive that of d(cos 6) from 
the identity cos 2 # = 1 — sin 2 0. 

2. From the identity sec 2 # = 1 -f- tan 2 #, derive the differential of 
sec 6. 

3. From the identity sin 2/9=2 sin 6 cos 6, derive another by tak- 
ing derivatives. cos 28 = cos 2 # — sin 2 0. 

Find the derivatives of the following functions : 

dy 

4. y •= 6 -f- sin 6 cos 6. — -= = 2 cos 2 #. 

— = cos 3 

dd ' 

dy 1 -f- cos 2 
dO ~ 2 (cos 0)* " 
^ jf cos x — sin x 

dy . 

-r- = s sin^;cosjf(smx— cos^;). 

dx ° ' 



5- 


jy = sin — J sin 3 0. 


6. 


sin 


J y(cos (9)" 




sin jv 


7- 


_> — ., ■ 


8. 


j/ = sin 2 2^c. 


9- 


y = sin 3 * -j- cos 3 .*. 


10. 


y —\ tan 3 — tan + 0. 


11. 


j/ s= -J tan 3 -}- tan 0. 



—=- = 2 sin 4Jtr. 
dx 



4= tan^. 

dd 

dd 



§ Vll 


.J EXAMPLES, 






e>7 


12. 


, y = sine*. 










— — = #* cos e x . 
dx 


13- 


jV = ^ sin .r 2 . 






^ 

^ 


= sin a: 2 -j- 2Jf 2 cos .a; 2 . 


14. 


j/ = # sin * 






dy 
dx 


= log « . a sin x cos jf. 


15- 


j> = tan 2 # -J- log (cos 2 #). 










d y *. sa' 
—— = 2 tan d fl. 

du 


16. 


y = log (tan (9 -j- sec 6*). 










dy n 


17. 


jr = log tan (Jar + £0). 










^ I 


dtf cos # 


18. 


^ = jf -f~ log COS ( \ 7t — 


x). 








^ 2 


dx ~~ 1 -|- tan x 


19. 


_y = log |/(sin -r) -(- log |/(cos jc). 






dy 

— — ■=. cot 2X* 

dx 


20. 


jf = sin «0 (sin 6) n . 




dy 

~dt> 


= » 


(sin 


(9) M - 1 sin(«4-i)(9. 


21. 


sin jtr 








dy 
dx 


cos 3 :r — sin 3 jc 


1 -\- tan Jt: 


(sin^-j-cos.^) 2 


22. 


y = e ax cosbx. 




dy 
dx 


= £< 


**(<2 cos <fo — 3 sin &*:)«, 




/a cos x — b sin .3; 






< 


— (2$ 


2 3* 


y a cos j; -J- sm ^r 


2 2 cos 2 x— b 2 sin 2 *" 


24. 


_>/ = e* (cos ;*: — sin X s ). 

1 
j/ = tan e x . 










-— = — 2e x sm x, 
dx 

1 1 

#V e* sec 2 £* 


25- 


^r a: 2 


26. 


y = e ax (a sin x — cos x). 






< 
< 


iy 
ix 


= (a 1 -f- 1 )e** sin jf. 


27. 


y = |/( 1 -(- sin x). 








dy 
dx 


= j|/(i — sin x\). 


28. 


(sin «jt) w 

y = — — . 

(cos mx) n 


dy 

dx 


mn (sin nx) m ~ l cos (m — n)x 
(cos w.r) w + 1 



68 FORMULA OF DIFFERENTIATION. [Ex. VII. 



29. 


y = tan |/( 1 — x), 






</>' _ — sec 2 4/(1 — a-) 
dx 2 |/( 1 — jf ) 


3°- 


y z= jf sin *. 




dx 


_ ^sin * / cos x j Q g jj, _j_ \ 


3 1 - 


y — sin (log nx). 




> 


<^r _ cos (log nx) 
dx x 

d y , ■ x 


3 2 - 


y = sin (sin x). 






-7- = cos ^ . cos (sin .%•) 
dx J ' 


33- 


2 


3 COS Jf 

sin^ 


+ 3 


x dy 2 


sin 2 .r cos x 


2 dx snr-rcos^Jt 



34. The crank of a small steam-engine is 1 foot in length, and 
revolves uniformly at the rate of two turns per second, the connecting 
rod being 5 ft. in length ; find the velocity per second of the piston 
when the crank makes an angle of 45 ° with the line of motion of the 
piston-rod; also when the angle is 135 , and when it is 90 . 

Solution : — 

Let a, b and x denote respectively the crank, the connecting-rod 
and the variable side of the triangle ; and let 6 denote the angle be- 
tween a and x. 

We easily deduce 

x — a cos 6 -j- \/{b 2 — a 2 sin 2 #) ; 
whence 

dx I a 2 sin 6 cos 6 \d6 

-W=-\ aSme+ W - * mto) It' 



In this case, — r- = Art. a = 1 and b = < m 

dt D 

'txn. a o dx 16^4/2 

When 6 = ak , — — = — ft. 

dt 7 

35. An elliptical cam revolves at the rate of two turns per second 
about a horizontal axis passing through one of the foci, and gives a 
reciprocating motion to a bar moving in vertical guides in a line with 



§ VII.] EXAMPLES. 69 

the centre of rotation : denoting by 6 the angle between the vertical 
and the major axis, find the velocity per second with which the bar is 
moving when 6 = 6o°, the eccentricity of the ellipse being j-, and the 
major semi-axis 9 inches. Also find the velocity when 6 = 90 °. 

The relation between 6 and the radius vector is expressed by the equa- 
tion 

a(i — e 2 ) 



r = 



1 — e cos 6 



dr 
When = 6o°, —— = — 1 2 4/3 n inches. 

36. Find an expression in terms of its azimuth for the rate at which 
the altitude of a star is increasing. 

Solution : — 

Let h denote the altitude and A the azimuth of the star, p its polar 
distance, / the hour angle, and L the latitude of the observer ; the 
formulae of spherical trigonometry give 

sin h = sin L cos/ -\- cosZ sin/ cos/ . . . (1) 
and 

sin / sin /= sin A cos h (2) 

Differentiating (1), p and L being constant, 

7 dh r . . . 

cos/£ -j- = — cosE, smp sin/, 
dt 

whence, substituting the value of sin _/ sin /, from equation ( 2 ), 

dh r ■ A 

-=j- = — cos L sin A . 

dt 

It follows that — — is greatest wnen sin A is numerically greatest; that 

is, when the star is on the prime vertical. In the case of a star that 
never reaches the prime vertical, the rate is greatest when A is greatest. 



7° 



FORMULA OF DIFFERENTIATION. [Art. 74. 




VIII. 

The Inverse Circular Functions. 

74. The graph of the function y = sin -1 .*- is given in 
Fig. 15; it is the same as that of sin x t Fig. 12, 
with x and y interchanged. It shows that the 
function is real only for values of x between 
-j- 1 and — 1, and that for any value of x be- 
tween these limits there are an infinite number 

^ of values of the function. To distinguish be- 

~ tween these, we have marked by the full line 

a portion of the curve corresponding to one 

value, and only one, for every admissible value 

of x. The values of y thus selected are all 
Fig i^ 

between ± \it and are called the primary values 

of sin -1 .*. The primary value of sin -1 .* may thus be re- 
garded as a one-valued function continuous for the range 
from x = — 1 to jt = -f 1. 

When not otherwise stated, the symbol sin - x x will be 
considered to mean the primary value. In particular, 
sin - \ — 1) = — 1 7r, sin -1 o = o, sin - x i =%7i \ Then, as proved 
in Trigonometry and illustrated by the figure, the other values 
of the inverse sine are all included in one or the other of the 
two expressions 

2n7t -f- sin - 1 x or (211 -f- i)tt — sin -1 .*, 

wnere n is a positive or negative integer. 

75. The graph of the function cos - x x is given in Fig. 16. 
The portion indicated by the full line corresponds to one 
and only one value of cos ~ 1 x for every admissible value of 



VIII.] 



THE INVERSE CIRCULAR FUNCTIONS. 



71 



x. These values are all between o and 7T, and are called the 
primary values of the function cos -1 ;r. Thus y 

the primary value may be taken as a one- 
valued continuous function for the range be- 
tween — 1 and + I- 

When not otherwise stated, cos -1 ;r will be 
taken to mean the primary value. The other 
values are included in the expressions 2n7t ± 
cos - x x, where n is an integer. In particular 




cos 



— 1 



(-1.)= n, 



cos 



- 1^ _ 1 



O = $7t, 



cos *i = o. 



Fig. 16. 



For negative as well as for positive values of x, the primary 
values of sin -1 ;? and cos -1 ;r satisfy the relation 



cos l x 



-1 n • -1 
1*? —sin x x. 

2 



iTT 



76. The graph of the function tan -1 ;? is the same as Fig. 

13 with x and y interchanged. 

The primary values correspond to 

that branch which passes through 

the origin as given in Fig. 17. 

Since all values of the tangent are 

possible, tan -1 ;? is a continuous 

Fig. 17. function for all values of x. The 

curve approaches the straight lines y == ± 2 7r as asymptotes. 

Thus, using the primary value, we write tan - x (— 00 ) = — -J^-, 

tan - x o == o, tan - 1 oo = \n. 

The other values of the inverse tangent are all included in 
the single expression 

nn + tan -1 ;tr, 



where n is a positive or negative integer. 



72 FORMULA OF DIFFERENTIA TION. [Art. TJ. 

77. It will be noticed that, for each of the functions 
sin -1 .*", cos _1 ^"and tan -1 ;r, the primary value is so taken that it 
is in the first quadrant (that is, between o and -J-7T) for all ad- 
missible positive values of x. Also, the primary value for 
negative values of x is so taken as to form a function con- 
tinuous for the whole range of admissible values of x, as shown 
by the graph in each case. 

In the case of the other three inverse circular functions, 
the same rule applies for positive values of x. It is not, how- 
ever, possible in the case of see -1 .*" and cosec -1 ^: to obtain a 
continuous function for positive and negative values of x, 
because these functions are not real for values of x between 
-|- I and — i. For cot -1 ^" we can. obtain a continuous function 
by taking the primary value between o and ?r, while x varies 
from -[-00 to — oo . 

The use of these last three functions can be avoided by 
means of the transformations: 

a _ ft _ a • m ft 

sec 1 -^-= cos l — . cosec * -# = sin" 1 —- , 

p a ' p a y 

a ' '_ /3 

cot -1 -77 = tan 1 — . 
p a 



Differentiation of the Inverse Sine. 

78. Proceeding in the usual manner for an inverse func* 
tion, let 

y = sin -1 .* - ; whence x = sin y. 

Then, by formula (d) ; 

_ dx 

dx = cosy ay ; whence dy = 



cos y 



§ VIII.] DIFFERENTIATION OF THE INVERSE SINE. 7$ 



Now cos y = ±4/(1 — sin 2 ^) = ± 4/(1 — x 2 ), but if y is re- 
stricted to the primary value of sin -1 ^, which lies between 
— \n and -|- k n i cos J ls positive. Hence, substituting, we 
have for the primary value 

dx 

<*(*"-■*)= V {i-*) w 

Accordingly, the primary value of sin _1 # is a continuous in- 
creasing function for the range of values of x between — 1 
and -f- 1, as shown in the graph, Fig. 15, p. 70. 
79. Similarly, when 

y = cos _1 x, x = cos y ; 

by formula (e), 

dx 



dx = — sin y dy ; whence dy = — 



sin;y 



Here sin y = ± 4/(1 — cos 2 )') = ± 4/(1 — # 2 ) ; but, when y is 
restricted to the primary value of cos" 1 ^ which lies between 
and 7t, sin y is positive. Therefore, for the primary value 
we have 

dx 

Accordingly, as shown in the graph, Fig. 16, p. 71, the 
primary value is a decreasing function. 

Differentiation of the Inverse Tangent, 

80. If y = tan -1 #, x = tan y, 

and, by formula (/"), 

dx=sec 2 ydy; whence rfy= 1- 



sec^ 



74 FORMULAE OF DIFFERENTIATION. [Art. 80. 

But sec 2 y = I -f- tan 2 }' = I -]- x 2 , therefore 

dx 
J(tan-^) = Yj~^' •-.... (w) 

Since I -f- ^ is always positive, tan _1 # is an increasing 
function as shown by the graph, Fig. 17, p. 71. 

The expression applies to all the values of tan -1 x, which 
in fact differ only by values of the constant nn in the expres- 
sion given in Art. 76. 

In like manner, or from the relation 





7t 

cot 1 x = — — tan _1 #, 


we derive 






7/ n dx 
dicot^x) = : — 5 

/ I _|_ x* 



(») 



Differentiation of the Inverse Secant, 

81. If y = sec _1 x, x = sec y, 

and, by formula (&), 

^ _ 

' sec y tan y 

where sec y = x and tan y = ± ^/(x 2 — 1). But, taking y in 
the first quadrant when x is positive, tany is positive, there- 
fore for the primary value 

_, dx 

J(sec-^) = xV(x? _ iy .... (0) 

In like manner, or from cosec" 1 ^ = i^r — sec _1 #, we derive 

dx 
ffcosec-'*) = - ^ _ , . . . (p) 



§ VIII.] DIFFERENTIATION OF THE INVERSE SECANT. 75 

in which the negative sign indicates that the primary value 
of cosec _1 x when x is positive is a decreasing function. 

82. To the formulae found above we add that correspond- 
ing to the versed-sine of Art. 72. Let 

y = vers _1 x, then x = vers y = 1 — cos y f 

dx 
dx = sin ydy, dy = -. — . 

* sin^ 

But sin y — 4/(1 —cos 2 )/) = |/[i — (1 — x) 2 ] = \/{2x — x 2 ) ; 

therefore 

dx 
d(vers -1 x) = —-7 — rr (a) 

y J \/{2X — X 2 ) W/ 

The inverse versed-sine is thus a continuous function for the 
range from x = o to x = 2, increasing from the value O to 
the value n. 

Homogeneous Forms of the Formula. 

83. In geometrical applications, the independent variable 
of an inverse circular function usually occurs in the form of 
the ratio of two lines, in accordance with the definitions of the 
direct circular functions. Putting x/a in place of x in the fore- 
going formulae, we have their homogeneous forms, in which 
each letter stands for the length of a line, the constant a 
taking the place of the unit of length. We thus obtain 

,/ . *x\ dx 

d[ sin - = 



a] \/((F — x z ) 

J t x\ adx 

(/tan" 1 - = 



a J ' ~ a 2 -f- x 2 ' 
\ a) ' " x \/{x 2 — a 2 Y 



j6 FORMULA OF DIFFERENTIATION. [Ex. VIII. 

Examples VIII. 

i. Derive d(sec~ 1 x ) from the equation sec - 1 x = cos- l -. 

(x\ x a 

cot -1 —) from the equation cot -1 - = tan -1 — . 
a) ax 

x -4- i dy i 

■2. y = sm - . 

4. y = sin - 1 (2^' 2 ). 

5. y = sin - ^cos .#). 

6. j/ = sin (cos - 1 a;). 
y. y = sin - *(tan x). 
8. _y == COS - ! (2 cos x). 
g. y = x sin - ^r -{-4/(1 — x 2 ). 

10. v = tan - 1 £*. 
■^ ^t; * * + e ~ *' 

dy _ 

11. v = (jr 2 -f- 1 ) tan" x x — x. — ==' 2X tan Y x. 

dx 

x dy 

12. y = <2 2 sin - * — I- .%V(tf 2 — -X" 2 ). — = 2 i/(<z 2 — .r 2 ) 

a dx x 

. mx dy m( 1 4- •^ 2 ) 

13. y = tan -1 





dx |/(l — 2JC — X 2 )' 




dy 4X 




dx 4/(1 — 4X*y 




dy 
dx 




dy x 




dx 4/(1 — j*: 2 )* 




</y sec 2 Jf 




dx ' |/(i — tan 2 jt")' 


*/y 


2 sin j; 


djf 


4/(1—4 COS 2 .* 1 )* 




— = sin x x. 
dx 




dy 1 



1 — X 2 ' dx I + (w 2 — 2)jf 2 4- ^ 

14. y = tan - x — ■?-£-. -f- = ■ ; = T J r . 

2 4- jt «:r 2 (jp* 4- x 4~ x ) 

jr dy 1 

15. _>/ = tan - * 



4/(1 — jt 2 ) </jf 4/(1 — ie 2 )' 

r _ _ 1 a <ty_ _ i_ 

x y - sec ^ _ ^. ^ _ ^ _ ^. 



VIII.] EXAMPLES. 77 

_ x dy a 

I7 ' y = Sm i /(x 2 -\-a 2 )' ^~ o^f^ 2 ' 

18. jf = sin" ^(sin .*). -~ = J |/(i + cosec .r). 



19. jy = 4/(1 — ^f 2 ) sin ^ — a:. 



dx 

dy x sin -1 .r 



2 V 



dx 4/(1 — ^ 2 ) 



20. jy = tan 



1 



_ m -j- x dy 

1 — mx' dx 1 -|- x 2 

I — Jf 2 <2?J/ 2 



21. J/ = COS — s. - „. 

i -j- .z 2 <£v 1 -}- •* 

— cos x dy _ _ 



22. jf = tan 1 |/ - 



1 -j- cos ^ dx 

..rsin -1 .r , . ,. „. dy sin -1 jr 



2* 



y(i — jr 2 ) <& (1- jf 2 ) 

yjf dv / x 

24. y — (x -\- a) tan - 1 a/ \/(ax). — = tan - 1 |/ — . 

, . , . ^ exp (sin - *x) 

25. y = exp (sm -1 x). — = — ?-± — - — -£. 
D ^ F v y <£tr 4/(1 — ^) 

26. j/ = exp [(1 -f- x 2 ) tan - ^x~\. 

— = (1 -|- 2jr tan - b;) exp [(1 -j- .r 2 ) tan - ^J. 

2 sin ~ x .%• , , 1 — x dy 2% sin - 1 x 

2 7- y — 7 ^T + log . -r- = " . 

V(l ~ ^) I + * <** (I - ^ 2 )I 

. _ x x tan or <^y <2 2 tan a 1 

|/(tf 2 — x 2 )' dx a 2 — x 2 ' \/(d 2 — x 2 sec 2 a)' 

_ ! / /^ 2 — x 2 \ dy _ xy(b 2 — a 2 ) 

29. J/ _ COS y \^— ^ • ^ - (d 2 _ x * )i/(a 2~ 



30. y = (1 - -* 2 )~* sin ty 



^ (£ 2 — x*)t/(a 2 — ^y 



^ 1 — j*: 2 1 -f~ 2Jf 2 

<£*• .%• Jt 2 

ax — 1 . , . 

3I '- y= 4/(i + ^) eXp(ata " *>• 



4/(1 — ^ 2 ) sin -1 jt;. 



dx (i-f.^2)l 

ZXf£ logarithmic differentiation. 



dy ( 1 -j- « 2 )jf 

exp (<z tan ~ l x). 



78 



FORMULAE OF DIFFERENTIATION. [Ex. VIII. 



32. y = tan -1 ^ -J- 4/(1 — j*; 2 )]. 

^ |/( 1 — x 2 ) — x 

llx " 24/(1 — x 2 )[i -f iy(i - x 2 )]' 

,b-+-acosx dy A/(a 2 — b 2 ) 

33. y = sm * r . — = *—i i m 

a -\- cos x dx a -f- b cos jt 

?4. v = sec -1 r— -*-= . — = 

2 yO 2 -f x — 1) </r* ^ +/(x 4 -\- x — 1) 

_ x 3^ 2 ^ — .X -3 dy 3a 

35* J' — *an — o p» -3— = -5 — : ^. 

<2° — 3^ ^ a 2 -f- .r 2 

36. j/ = tan -1 -^ = — . — = — — £-..• 

# -j- <? cos x <£r b -J- <z cos :r 

,7 v = sin- * *V( a -' d ) ±_ _. V( fl ~ b ) 

4/[a(i -for 2 )]* <&: (i -f JC 2 ) |/(> 4- fo; 2 )* 

jv 3 — 2 </v ? 



38. v = cos 



— 1 



a; 3 dx x ^/(x s — 1 )* 

</v JV 2 I jx 1 I 

39. >> = x exp (tan - 1 ^r) — = — ^ exp (tan -1 ^:). 



IX. 

Recapitulation of Formula. 

84-. We have now obtained the formulae required in Art. 
42 to enable us to differentiate all the functions which can be 
expressed by combining the elementary functional symbols 
and the algebraic operations. They are here recapitulated : 

d(x + y + z+...) = dx + dy + dz+ (A) 

CL\0C-,0Ca ... OCpj — %2 ' • • OCmj/OC-i r~X-iXo • • • Xp Q/Xn I ... • V / 

(x\ ydx—xdy 

%r~T~ (C) 



IX.] RECAPITULATION OF FORMULA. 79 

d(x n ) = nx n ~ x dx (a) 

dx 

d( - los * x)= ^b <*> 

dx 

d(Xog x) = - {V) 

d(a x ) = log a . a x dx (c) 

d{e x ) = e^Jx (V) 

(/(sin 0) = cos ddd ((/) 

(/(cos 6) — — sin dO (e) 

d6 
d(tan0)= — ^^szcWdd. ...... (a) 

COS P * ' 

7/3 

rf(cot d) = — -7-yw = — cosccWdd. ... (#) 

sin Odd 
(/(sec 0) = — - = sec 6 tan dO. „ . . (&) 

cos 2 # v 

7/ \ cos #^# 

(/(cosec 0) = . 9 - = — cosec # cot 000. . (i) 

sin 4 !? v ' 

(/(vers 0) = sin dd (J) 

dx 
rf(sin-*) = V(I _^ • (*) 

„ , x (/X 

flees-*) = - ^ _ ^ (Q 

(/jc 

d(tan-'*) = ■ (m) 

dx 
d(cot~ l x) = - y-p^ (n) 

dx 
rf(sec-%) = - ^ _ i} („) 

7, « \ ^ 

(/(cosec -1 ^) = — — — ^ r. ...... (/>) 

v ' x i/(x 2 — i) KrJ 



SO FORMULAE OF DIFFERENTIA TION. [Art. 84. 

d(vers - V) = — — ; -r (q) 

v J 4/(23; — x z ) w/ 

Differential of a Function of Two Variables. 

85. Formulae (a) . . . (q) express the rate of variation in a 
function due to a known rate of variation in a single variable 
upon which it depends, by means of the differentials which 
measure these rates. 

This they do in each case by giving the " differential co- 
efficient," or derivative which measures the relative rate. 

The formulae (A), (B) and (C) have hitherto been used 
in the combination of variables which were themselves func- 
tions of a single independent variable. But in them the 
several variables may be independent, so that they express 
the rate of a function of several variables in terms of the 
values of the variables and their rates. 

Now the form of these equations is such that the differen- 
tial of the function is the sum of several terms containing 
respectively as factors the differentials of the several variables. 

These several terms are called partial differentials, and 
their coefficients, partial differential coefficients. In contradis 
tinction, the differential of the function of several variables is 
called the total differential. 

86. The same thing is obviously true with respect to the 
form of differentials resulting from the substitution of formulae 
(A), (B) and (C) in the other formulae. For example, sub- 
stituting from (C) in (k), (m) and (o) respectively, we find 

ydx — xdy 
/ . x\ y* ydx - xdy 



§ IX.] FUNCTIONS OF TWO VARIABLES. 8 1 



<(*»-f)= 

d[ sec -1 — ) -. 

\ y) 


ydx— xdy 

y 2 ydx — xdy 


• (2) 

• (3) 


;ydx — aprfy 

y 2 ydx — #dy 


~ x I lx l \~~ x \/{x 2 — y*y 

yV\f 7 



Each of these total differentials consists of two parts, one 
containing dx as a factor and the other dy. The coefficients 
in these partial differentials are also called the partial deriva- 
tives of the function of x and y. This principle, namely, that 
the total differential is simply the sum of partial differentials* 
each of which vanishes when the corresponding variable is 
made constant, shows that each partial derivative can be 
found by regarding the function as a function of the cor- 
responding variable alone. Thus the partial derivative which 
is the coefficient of dx is nothing more than the derivative 
with respect to x. 

Accordingly, the coefficient of dy in equation (2) above, or 

x 
partial derivative for y of tan -1 - is simply the derivative of 

y 

X V 

tan -1 —, or of cot -1 —, with respect to y. In like manner, in 

y x 

equation (3), the coefficient of dy is the derivative with respect 

y 

to y of cos -1 — . 
x 

* We do not at present need to show that this is true of every conceivable 
function capable of differentiation since we are dealing only with the functions 
expressible by the elementary symbols. 



82 FORMULA OF DIFFERENTIATION. [Art. 87. 



The Derivatives of Implicit Fttnctions, 

87. When y is a function of x given in the implicit form, 
the relation connecting the variables is the result of equating 
to zero a certain function of x and y. The result of differ- 
entiating the relation between x and y is therefore equivalent 
to equating to zero the total differential of this function. 
This gives a relation between x, y, dx and dy, by means of 
which the ratio dy/dx can be expressed in terms of x and y. 
For example, taking the illustration of an implicit function 
given in Art. 5, namely 

ax 2 — ^axy -j- y B — a 3 = o, . . . . (1) 
differentiation gives 

(2ax — ^ay)dx — (^ax —3y 2 )dy =0, 
whence 

dy __ a(2x ~sy) 



dx ^(ax — y 2 )' 



(2) 



88. This equation gives the derivative of the implicit 
function v, not directly as a function of x, but implicitly so, 
by virtue of the original equation. It determines the value of 
the derivative for any known simultaneous values of x and y. 
Thus, in the illustration above, if we put y = a in equation 
(1), we obtain x = o or x = 3a. Hence (o, a) and(3#, a) rep- 
resent simultaneous values of x and y. Denoting the corre- 
sponding special values of the derivative by suffixes, we find 
from equation (2), by substitution, 

dy~] , dy~] 1 

— =1 and — = -. 

dx_j 0> a dx_\ 3 a t a 2 



§ IX.] EXAMPLES. 83 

Regarding (1) as the rectangular equation of a curve, we 
have thus determined two points on the curve ; and also, for 
each of them, the value of the gradient at that point. 

Examples IX. 

Find the total derivatives of the following functions : 

1. u = xye x + 2y . du = e x y [y(i -\- x)dx -{- x(i -\- 2y)dy~\. 

x , ydx — xdy 

2 . u = log tan — . du = 2 -. 

y 2 sin 2 - 



y 



. x 7 ydx — xdy 

3. u = log tan -1 —. </# = 



fifo 



y „ .* 

(^2 _|_j/ 2 ) tan -1 — 

4. « = : . 

x -\- y 
\y — x—2 ^{xy*)\ 0dx -\-[x — y — 2 \/(xy)~] \/xdy 
2 |/(xy) (>~ +yf 

e*y . (x 2 4- y 2 — x)ydx 4- x 2 dy 

5 . u — r du = e x — - — -. 

(x 2 -\- y 2 Y ( x2 ~\~ y 2 )* 

•, x— y _ ydx — xdy 

6. u = tan - x — -^. </« = - — f-- 

x -\-y x 4 -\-y l 

, x 2 — y 2 2xy (ydx — xdy) 



jx~ — r , 

7. « — 4/—- =. #« = 

' y x l 4- v 2 



o. u = lo^r 7 — d 5\» du — — t ^— -- 

.%■ — ^/(^ — y 4 ) y /^{x~ — y 4 ) 

9. From x = r cos 6 and_y = r sin 6, deduce 
(dxf -f (^) 2 = (^) 2 + r\ddf. 

10. Given ^ = r cos and y — r sin 0, r and being indepen- 
dent variables, prove that 

dy sin 6 -\- dx cos 6 — dr 
and 

djy cos 6 — dx sin 6 = r</#. 



8 4 



FORMULA OF DIFFERENTIATION. [Ex. IX. 



ii. Given x = r cos 6 and y = r sin 6 ; eliminate 6 and find dr ; 
also eliminate r and find d6. 

xdx -4-ydy ■ xdy — ydx 

dr — — — - — , and do = ~ , „ — 

v(. x * +y ) *■ +y 

12. \iy is defined as an implicit function by the equation 

xy 2 .-\- x 2 y — 2 = o, 
find the values of its derivative corresponding to x = i. 






= — i, and -7- 
dx 



= o. 



J 1,-2 



13. Given x 2 (y— 1) - r -J /3 (^+ I ) — r ; nn ^ an expression in terms 

dy 



of x andy for ^-, and also its numerical values when_y = 2. 
dx 



dy 



= — 6, and 



^y~ 



_l — 7, 2 



6 
2 3* 



a^ J_l, 
14. Show that the equation 

xy z — $x 2 y -}-- 6y 2 -f- 2X = o, 
is satisfied by (2, 1) and by (o, o); and find the corresponding values 



of 



dy 

dx 

1 x — a y — a 

1 5 . tan ~~ tan 



dy' 
dx_j2, 



x -\- a 
16. y = 1 + xeV - 



y + a 



= b. 



17. (x —y)j n = •*+->'• 

18 . („v 2 +y 2 ) 2 = a 2 x 2 - b 2 y 2 . 

19. ye ny = ax m . 

20. y z — $y sin ~~ x x -\- x 5 = o. 



3. ^1 

2 ; ^Jo,o 

</y y 2 -\- a 2 
dx x 2 -f- a 2 ' 
^ e* 

<sfa; 2 — j/ 
<^v 2y 2 

dx ixy — n{x 2 — y 2 ) 
dy [a 2 — 2(x 2 -K> ;2 )]" r 
dx [b 2 -j- 2(x 2 -\-y' 2 )]y 
dy my 

dx x ( 1 -|- ny) 

dy y — x 2 ( 1 — Jf 2 )2 

— = 3^ — i- 

dx (zjy 3 — x *) i 1 — x 2 )^ 



\ IX.] EXAMPLES. 85 

dy ny . 

21. y sin «jr — ae nx+y = o. -— = ( 1 — cot nx). 

dx 1 — _y ' 

22. _y tan *# — y + _r 2 =0. -5- —- — - — , —-£. 

dx (1 -{-xr)(y z -\-x z ) 

Miscellaneous Examples. 

x } dy x -j- 2. 

|/(l + .*) ^ 2(1 +^)t 

/« 2 — jr 2 ay (<z 2 — b 2 )x 

I 2 ' y ~Vb 2 — x* ~dx~ {a 2 -x"f{h l -x 2 f 

\/{a -\- x) dy ^/a{ \/x — tfa) 



3- y 



y'a -j- j^x dx 2 \/x \/{a -\- x) ( ^/a -\- y'x) 



4. y = (^/x — 2 \/a) |/(|/a -f- \/x). 



s-y = 



dx 4\/ (V a ~\~ V x ) 
[x — i)(e*.-\- i)e* dy e*{xe' 2x — 2xe x -j- 2e x — x) 

e* — 1 <£c ~ (e* — 1 ) 2 

( 1 + x 2 )i , ^ x 

/i + x\i , dy x s 

i-y = l °s L— J ~ * tan * & = r^ 4- 



x dy 1 / (x A- a 
8. y = log [, + **» - -»)] + sec - _• -£ = - / (^ 

^ — ^ , dy 8x 2 

y (x 2 -{- iy dx (x 4 -\- i) 6 

a + \Z(d 2 — x 2 ) 
10. y = a log — ! — — ' — ^(a 2 — x 4 ). 

dy \/{a 2 — x 2 ) 

dx x 

(1+ $x + 3X 2 )h dy (1 + xf 



11. y 



x 



i 2 .j= a \og a+ * /{a *- X * ) - i /(a>- X >). 



dx x\i -j- t>x -\-$x 2 )* 



x 

dy \/(a 2 — x 2 ) 

dx x 



B6 



FORMULA OF DIFFERENTIATION. [Ex. IX. 



1 3- 

14. 

16. 
17. 



/ T — COS X 

y = log 4/ — ■ • 

r I -j- COS JV 

/a — b x' 

V — rx' tan_ 



^/ 



to« — 1 



jy = tan 



y = sec _1 



2^ 4 — I 

2W -r 



J/ = cos _1 



y =z a cos 



X 



X in + I 

, a — x 



dx sin x 
dy __ tf(a* — b 2 ) 
dx 2 (a -[- b cos x) 
dy _ 2 

dx~ 4/(1— #*)" 

^ 27ZJtr w_1 

</.r — jr aw -|- 1 ' 






.r 



jy = cos ~ 1 x 



A 



I — X 



19. 



20. 



y = x 
y = x 



■y+x 



dx ' \f[b 2 — (a — x) 2 ] 
dy _ 4/(1 — x) 
+ ■* ' <** ( 1 + x)i " 

<§/ j^+^jr y+x - 1 4- log Jf 4- 1 ) 
^t; ~ 1 — x y+x log or 



dx 



— (wlog Jtr -f- 1 )x* n x"- 1 . 



2 1 . y = x x * 



— z=X x X* 

dx 



(log x) 2 -j- log ^ -J 



x 



J 



22. 

2 3- 

24. 

2 5- 
26. 

27. 



^ dy exp (x tan /?) cos x 

y = exp (^ tan p) sin (^ 4-/5). -7- = — 



_y = log[^+^(i+^*)]. 



cos /? 



j/ = tan 



dy 



tf(b — a) 



y = 



\/(b — a) 
<£r 2(14- x) 4/(0 4- &%•) 
^ 2 

*&; sin 3 jtr* 
_ 1 Jtr 4/2 ^ 44/2 

1 — jr 4/2 4- x l ' 1 — jv 2 <2^ 14- -V 4 ' 

I /i I2S . 65^ l^ 2 \ ^ 

|_4_ 4^_4_ 5 ^3j + 5 log—-—. 
3 2 / i -}-^ 

dy 1 

</.%• jf 2 ( 1 4- x) 5% 



x cos x 

y = log tan — — . 

2 sm^ 

y = l °sz — - ,„ ' ^ + 2 tan 



(1 + ^) 



§ IX.] EXAMPLES, 87 

1 -[- x , n 1 -\-x 4- x 2 , x a/t. 

28 . y = log^ + i log ,1^ + V3 tan-^ 



<£r I — Jr 6 

m 

29. y = (1 + ^ 2 ) T sin {m tan _1 .a?). 

; w— 1 

uV 

— = m(i 4- x 2 ) 2 cos [(m — 1) tan~~Vl. 

(.%•— i) 2 .2^4-1 cV 6 

30. j/ = log -\- — r 2 V3 tan - 



jr 2 -(-.%■-{- 1 4/3 */.# .a; 3 — 1 

ow 
dy 



p x I — x 

21. If y = — — , show that 



32. Given « = xs -j - a s ^ n z-\-az cos 2, and x -=.a — a cos ; 
prove that 

du (2a — x\z 



dx 



C-^T 



CHAPTER III. 

Successive Derivatives. 



X. 

Velocity and Acceleration. 

89. We have in Art. 17 employed the velocity of a mov- 
ing point to illustrate the rate of a variable, the variable x 
being represented by the distance of the moving point from a 
fixed origin in the line of motion. If we now represent this 
velocity by v, we have 

dx 

v = w ( J) 

When this velocity is variable its rate of variation is the rate 
of the rate of x. Since dt is constant, we have, by differentiat- 
ing equation (1), 

_ d(dx) dv d(dx) 

dv = -*-> whence * = w • (2) 

The rate of the velocity is called the acceleration of the mov- 
ing point, and may be denoted by the single letter a. In 
equation (2), d(dx) is generally written in the abbreviated 



§ X.] VELOCITY AND ACCELERATION. 89 

form d 2 x, which may be read " ^-second " x\ also the marks 
of parenthesis are omitted in the denominator. Thus we 

write 

dv d 2 x 

a = di=dfi (3) 

90. In equations (1) and (3), x, the space described, is a 
definite function of t\ v, the velocity, is the derivative of the 
space with respect to / ; while «, the acceleration, is the 
derivative of v with respect to t, and is called the second 
derivative of x with respect to t. 

Just as a positive value of the first derivative v indicates 
algebraic increase of x, so a positive value of the second 
derivative a indicates algebraic increase of v. The term 
acceleration is of course derived from the case when both the 
velocity and its rate are positive, so that the moving point is 
hastened, A negative acceleration is a retardation of a posi- 
tive velocity, but an algebraic increase of a negative one. 

91. For example, suppose it to be known thmt the space 
described in the time t by a freely falling body varies as the 
square of the time, so that it may be represented by 

x = \gt\ 

where g is a positive constant. From this we derive 



dx 

v = 

and 



v=z Tt=^ 



dv d 2 x 

a = dt = ^F = £- 



In this case, therefore, the acceleration is constant and posi- 
tive. Accordingly, the velocity which is positive for positive 



90 SUCCESSIVE DERIVATIVES. [Art. 91. 

values of /, is increasing. At the instant when t = o we 
have x = o and z; = o. Supposing the body to be already in 
motion before that instant, we see that v is negative for nega- 
tive values of /, that is, the body was moving in the oppo- 
site (or upward) direction, and then the positive acceleration 
implied a decrease in the negative velocity. 

92. The time, space, velocity and acceleration may be 
regarded as four variables connected by the two general dif- 
ferential relations 

dx dv 

v = dt< a = T? 

Therefore, when one other relation between them is given, 
three of the four variables become definite functions of a 
single independent variable, which may be any one of the 
four. The problem of so expressing them under different 
forms of the additional datum relation (which makes the 
motion definite) is the application of the Calculus to the subject 
of rectilinear motion, and for the most part requires the inverse 
process of Integration. 

We may here, however, notice another general differential 
relation found by eliminating dt from the two given above, 
Thus 

dv dv dx dv 1 d(v 2 ) , x 

a= — =-—•-- = v-r = — V^ ; . . . (1) 

dt dx dt dx 2 dx v ' 

that is to say, the acceleration is always equal to the space- 
derivative of one-half the squared velocity. 

By means of this we can, when v is given in terms of x, 
express a in terms of x. For example, if we are given 
v = n \/(x 2 — a 2 ), we thus find a = n 2 x. 



§ 



$X.l 



COMPONENT VELOCITIES. 



9* 



Component Velocities and Accelerations. 

93, When a point moves in a plane curve its motion is 
most conveniently discussed by means of points geomet- 
rically connected with it which have rectilinear motion. Refer- 
ring the path of the point P to rectangular coordinates as in 
Fig. 18, these points are the pro- 
jections R and S of P upon the 
axes. Their velocities (which are 
the rates of x and y respectively) 
are called the component velocities 
of P. The actual velocity of P is 
the rate of s, the space described 
measured along the curve from 
some fixed point of it, as A in the 
figure. Denoting it by v, and 
the component velocities by v x and v y , we have 

dx 




Fig. 18. 



V = — 



ds 

It 



v„ = 



dt' 



dy 
y dt 



94. In the differential triangle, constructed as in Art. 37, 
PP' represents ds and P'PB is 0, the angle of slope. Then, 
since the triangle is right-angled, we have 

dx= cos ds, dy = sin <P ds 

and 

ds 2 = dx 2 -{- &f' 

Dividing by dt and df respectively, we find 

v x — v cos cp, v y = v sin 
and 



v 2 = v A 



v. 



■y . 



92 SUCCESSIVE DERIVATIVES. [Art. 94. 

These equations serve to determine the actual velocity, v, 
and the slope of the curve, when the component velocities 
are given. 

In these equations, is the angle of inclination of the 
actual motion of P, thus distinguishing between the two 
values of which in Art. 38 correspond to the same gradient. 

95. The accelerations of R and S may be denoted by 
ol x and oi y \ thus, 

d?x (Py 

ex — — ex — — 

* ~ dP y ~~ dt 2 ' 

These are called the component accelerations of the point P. 
It is evident that their values determine not only the accel- 
eration of the point P in its path, but the curvature of this 
path. 

Examples X. 

1. The space in feet described in the time / by a point moving in 
a straight line is expressed by the formula 

x = 48/ — 16/ 2 ; 

find the acceleration, and the velocity at the end of 2 -J- seconds; also 
find the value of / for which v = o. 

a = — 32; v = o, when / = 1 J. 

2. If the space described in / seconds be expressed by the formula 

4 
x = 10 log , 

4 + ^ 

find the velocity and acceleration at the end of 1 second and at the 
end of 16 seconds. When / = 1, v = — 2 and a = f. 

3. If a point moves in a fixed path so that 

show that the acceleration is negative and proportional to the cube of 
the velocity. Find the value of the acceleration at the end of one 
second and at the end of nine seconds. — \; — T -J-g. 



X.] EXAMPLES. 93 

4. If a point move in a straight line so that 

x = a cos \nt. 



show that 



a = —±7i 2 x. 



5. If :r = ae t -f- fo""', 

prove that 

a = ^. 

6. If a point move so that v = \/{2gx), determine the acceleration. 
Use equation (1), Art. 92. a = g. 

7. If a point move so that we have 

v % = c — jj. log x, 
determine the acceleration. a = . 



2X 



8. If a point move so that we have 

2)1 



v 2 = c + 



JUX 



^2 + py 

determine the acceleration. a =■ 

(xt + py 

9. The velocity of a point is inversely proportional to the square 
of its distance from a fixed point of the straight line in which it moves, 
the velocity being 2 feet per second when the distance is 6 inches ; 
determine the acceleration at the distance s feet from the fixed point. 

= feet. 

2 s 

10. The velocity of a point moving in a straight line is m times its 
distance from a fixed point at the perpendicular distance a from the 
straight line ; determine the acceleration at the distance x from the 
foot of the perpendicular a = m 2 x. 

11. The relation between x and / being expressed by 

/211 . 9N t . 2X 

lA/ — = a/ (ax — x') — -itf vers 1 - — ; 

y a v v ' & a > 

find the acceleration in terms of x. a = „. 

x z 



94 SUCCESSIVE DERIVATIVES. [Ex. X. 

ID 

12. If v 1 = A + — , show that the acceleration varies inversely as 

x 

the square of the distance from a fixed point in the line of motion. 

13. If v 1 = A -\- Bx -f Cx 2 , show that the acceleration varies as 
the distance from a fixed point in the line of motion. 

14. In " tram motion " each end of a rod AB is constrained to 
move in one of two grooves crossing each other at right angles at O. 
If the velocity of one end is proportional to the distance of the other 
end from O, prove that its acceleration is proportional to its own 
distance from 0. 

15. A point referred to rectangular coordinate axes moves so 
that 

x = a cos / -f- S, y = a sin / -f- c ; 

show that the velocity is constant and that <fi uniformly increases. 
Find also the equation of the path described. 

16. A projectile moves in the parabola whose equation is 

y = x tan a — -„ „— x 4 

(the axis ofy being vertical) with the uniform horizontal velocity 

v x = V cos a ; 

find the velocity in the curve, and the vertical acceleration. 

v = ^(V* — 2gy); tx y = — g. 

17. A point moves in the curve, whose equation is 

2. 2 s 

xz -j-j/s" =z a$ f 

in such a manner that v x is constant and equal to k ; find the acceler- 
ation in the direction of the axis of y. asfi* 



18. A point moves in the hyperbola 

fi = p 2 x* -j- <p 



y — 4i* 

3^3^3 



§ X.] EXAMPLES. 95 

in such a manner that v x has the constant value c; prove that 

Vy — P L , 

p 2 c 2 g 2 
and thence derive a y . ot y = — g— . 

19. A point describes the conic section 

y 2 = 2tnx -j- nx 2 , 

v x having the constant value c\ determine the value of a y in terms oiy. 

m 2 c 2 



XL 

Successive Derivatives of a Function. 

96. The derivative of f(x) is another function of x, which 
we have denoted by f ; (x); if we take the derivative of the 
latter, we obtain still another function of x, which is called the 
second derivative of the original function f{x), and is denoted 
by /""(#). Thus if 

f(x) = x 3 , f{x) = 3X 2 and f"{po) = 6x. 

Similarly the derivative oif"{pc) is denoted by f'\x) y and 
is called the third derivative of f(x) ; etc. When one of these 
successive derivatives has a constant value, the next and all 
succeeding derivatives evidently vanish. Thus, in the above 
example, f'"{x) = 6, consequently, in this case, f lv {x) and 
all higher derivatives vanish. 

97. When the function is denoted by the single letter y 

d 

we have seen in Art. 35 that ~r~ may be taken as the symln 



g6 successive deriva tives. [Art. 97. 

of the operation of taking the derivative. The single letter D 
is often used for the same purpose, and an exponent is applied 
to the symbol to denote repetitions of the operation ; thus 
D{Dy) or D 2 y is the second derivative of y, and D n y is the nth. 
derivative. In like manner the higher derivatives may be 
denoted by 

<A 2 / dV /d 



dx) y ' \dx) y * ' ' ' \dx) y 

in which the independent variable is directly expressed. 
These last symbols are more usually written in the abbrevi- 
ated forms 

d 2 y d s y d n y 

dx v dx iJ ' ' ' dx ni 

although the former symbols are the more accurate, because 
the operation to be performed n times is that of differentiating 
and removing the factor dx after each differentiation* 

Geometrical Meaning of the Second Derivative, 

98. When the graph of the function y =f(x) is drawn, we 
have seen that 

£=/'{*)= tan <p, 

<p being the inclination of the curve to the axis of x; hence 
d 2 y ff d (tan cp) 

dx^ -f"w = ~~~ax * 

* It is to be noticed that, on this account, it is immaterial whether dx is constant 
or variable. But when dx can be assumed constant (like dt in Art. 89), we may 
suppose all the differentiations performed first, and (dx) n removed by division 
afterward. 




§ XL] THE SECOND DERIVA TIVE. 97 

If now we suppose a point to describe the curve in such a way 
that the rate of x is constant and positive, the value of the 
second derivative gives the rate at y 
which tan 0, the gradient of the 
curve, varies. In Fig. 19 are shown 
several curves having a common 
tangent, MN, and a common point 
of contact at C, so that the value of 
the functions represented and also_n_ 
of their first derivatives are equal 
at C. But it is obvious that in the Fig. 19. 

curve AB the gradient is increasing more rapidly than it is in 
the curve AB' , which lies nearer the tangent and is therefore 
said to have less curvature. Thus the value of the second de- 
rivative at C is greater for the curve AB than for the curve AB' . 
Again, for the tangent itself, which represents a linear func- 

d 2 y 
tion, the value of -r-^ is zero; while for the curve A 'B" (in 

which the gradient is decreasing as x increases) its value is 
negative. 

Accordingly, when the second derivative is positive, the 
curve lies like AB above the tangent line and is concave as 
viewed from above; and, when the second derivative is negative, 
it lies below the tangent and is convex as viewed from above. 

99. When a curve crosses the tangent line at the point of 
contact, in which case that point separates a convex from 
a concave portion of the curve, as in Fig. 20, the point is 
called a point of inflexion or of contrary flexure. Suppose 
the point of contact, carry i?ig the tangent with it, to 
move in the positive direction along the curve. As it passes 
through a point of inflexion changes from a state of decreas- 
ing to a state of increasing, as in the figure, or vice versa. 




98 SUCCESSIVE DEE IV A TIVES. [Art. 99. 

The tangent at the point of inflexion is called a stationary 
tangent, because after turning in one direction it stops and 
y „ then begins to turn in the 

n opposite direction. The value 
of the second derivative there- 
fore changes sign as x passes 
through a certain value; 
hence, if it is a continuous 
function of x, it must take the 
value zero. Hence, to find 
the abscissa of a point of in- 
FlG - 2 °' flexion, we put the second 

derivative equal to zero, and if the equation so formed has a 
root for which the function is real we must then ascertain 
whether the second derivative changes sign. 

d 2 y 
For example, the equation y = x gives -=-% = 6x, which 

Q/X 

vanishes when x = o; the curve has a point of inflexion at the 

origin because 6x changes sign as x passes through zero. 

d?y 
Again, y = x i gives -=-^ = i2x 3 , which also vanishes when 

#=0; but, since it does not change sign, there is, in this 
case, no point of inflexion. 

Higher Derivatives of Implicit Functions. 

100. When y is given as an implicit function of x, the 
higher derivatives, like the first derivative (Art. 87), can in 
general be found only in terms of x and y ; hence the numerical 
values of these derivatives can be determined only for known 
simultaneous values of x and y. The following examples will 
serve to illustrate the method of finding such derivatives. 



.u *v ... 



§ XL] HIGHER DERIVATIVES. 99 

Given 

log(x + y) = x — y\ (i) 

we obtain, by differentiating and reducing, 

(x -f- y -\- i)dy + (i — x — y)dx = o; . . (2) 
whence 

dy x -\- y — 1 

dx x + y + 1 " 

Differentiating, and dividing by dx, 



(3) 



cpy (* + y+o(* + ^-(*+y-o('+|) 

(foe 8 (*-f:y+i) a 

Jv 
substituting the value of -=-, we obtain 

<Py = 4(x + v) 

<fce* (x + v + 1) 3 W 

In like manner, the third derivative may be found. 

Simultaneous values of x and y are readily found in this 
case. Thus, if we put x -f- y = 1, we have x — y = o, whence 
x = \ and y = i; by substituting these values in equations 
(3) and (4) we obtain 



dy' 



= o and —4 



1 



dxA\,\ dxtJ^, $ 2 

These results show that the curve represented by equation 
(1) passes through the point (£, £), is there parallel to the axis 
of x and lies above the tangent line. 



LofC. 



100 SUCCESSIVE DERIVATIVES. [Ex. XL 



Examples XI. 

I. \if{x) = L±^ } find/ v (*)- f*(x) = 



X (i — x) 6 ' 

Tr ,/ v a c a run \ snn \ n(n -\- i) (n-f- 2)a 

2. U/(x) = -, find/"»- /'"(*) = ~ -* ^ s ^ ■ 

3. If j|/ is a function of .* of the form 

Ax n + .fr*"" 1 -f . . .+ Mx + N, 
prove that 

d n y 



dx n 



n\ A. 



4. If/O) = b ax , find/ v (^). f y (x) = a\\ogbfb ax . 

5. If/(-r) = x* log (flz*), nnd/ IV (*). f lv (x)= - . 

6. Iff(x) = log sin *, find /"'(•*). /""'(■*) = 2 C ° S X , 

7. If/ (a) = sec x, find /"(.*) and/'"(*). 

f'{x) = 2 sec 3 .* — sec x\f n {x) = sec .* tan ^(6 sec 2 .* — 1), 

8. If/(*) = tan x, find /'"(.*) and/ IV (.*). 

f'"{x) = 6 sec 4 .* — 4 sec 2 * ; ./" IV (-*0 = 8 tan * sec 2 .*(3 sec 2 * — 1), 

9. If/(*) = #*, find/"(*). /"(■*) = ■*•*•( 1 + log *) 2 -f **-*, 

J. 6^y <2^ 3 y 1 i_ 

10. Ify = e*, nnd— 3 . — = - ^-(1 + 6x + 6x 2 )e* , 

11. If y = e~ x find Z^. Z^ = 4^(3 — 2x 2 )e~ x , 

12. Ifj/= log (** + *-*), find -r^. -^=—8 ; ■ , 3 , 

dx 3 dx 3 (e x -j- <?-*) 3 

dx* = (e x — i) 3 ' dxF~~~ (e* — i) 5 

14. If 1/ = sin- 1 .*, find Zfy. D'y = 9 * + 6 % \ 

(1 — x 2 y 



if j * A d y a d y 

13. If y == . find -r-5- and -j—r- 

-^ e*— 1' a 7 .* 2 </* 4 



XL] EXAMPLES. 101 

15. \iy— e sinx , find D 3 y. 

jy>y = — £Sin x CQS x s j n x ^g^ n x _j_ ^ 

.* ^ -, ^ <Py 1 —log x 

16. If _y = - , find 



1 -|- log x ' dx 2 ' dx 2 x(i -f- log x)* 

17. If y = (cos -1 ;r) 2 , show that the following relation exists be- 
tween its derivatives : 

d 2 y dy 

(1 — x 2 ) -~ — x—— = 2. 
dx* dx 

18. \iy = a cos log x -\- b log sin x, show that 

19. If xy = ae x 4- <fe - *, show that 

20. If jy = ^4tf* sin [x + **), show that 

dh dy 

2 -^— -4- 2V — O. 

21. Find the value of T^-(sin 6), 6 being a known function of/. 

^ , . n. » fdOV . „dS d*0 n d?d 

df* 



^(sin 6) = -cos ^_j -3 sin 6> - * ^ + cos _. 



cfiu 
22. Ifj> is a function of x, and if u = y 2 logj^; find -j— 2 



% = ^ l ° SJ, + 3) (|)+X* log-'' + 1) g. 



23. If u = — , y being a function of ^, find -— . 
y dx 2 



(Pu 2 dy 2X / dy\ 2 x d 2 y 

dx 2 y 2 dx * y 3 \dxj y 2 dx 2 ' 



102 SUCCESSIVE DERIVATIVES. [Ex. XI. 

24. Distinguish the concave from the convex portions of the 
curves^ = sec x, y — tan x and y = sin x. 

25. Find the point of inflexion of the curve 

y — 2 X* — $X 2 — \2X -f 6. (£, — |). 

26. What portion of the curve 

y = jr 4 — 2a: 3 — 1 2jt 2 -\-wx -\- 24 
is convex? Between (2, — 2) and (— 1, 4). 

27. Show geometrically that at a point of a curve where the gradi- 

d y d 2 x 

ent is positive -— ~ and — r— ■ have opposite signs. - 
ax* dy l 

Consider the position of the curve relatively to the tangent line, as in 

Art. 98. 

28. Determine —7- when y = fix). — — = — r *\ ' 

29. Find the value of — ^ for the curve and points considered in 

Arts. 87 and ^. 

d 2 y~\ 2 d 2 y~ 

dx Jo,a " 3^ ' ^ 2 . 

30. Show for the same curve that it lies above the tangent at the 
point ( — a, o) and below it the point (a, o), and that the curvature is 
the same at these two points. 

Si- I*> - 1 -** = o, find %. % = 7 ^— y . eX 



3 a,a 12^ 



fl^ 2 djf 2 (2 — J/) 3 

/ 3 y ^_ 2(5+8^4 

d 3 y d s y 2\x 



/ , v j. ^ ^ ^ 2 (5 + 8y 2 + 3/) 

32. If, = tan (, +y), find ^ . ^ = - ^ y ^ ^ ' , 

33. Ifj/2_|_j, — ^ find 



d^ 3 " </jc 3 (1 -f- 2y) 5 ' 

4- v, find - 
they satisfy the relation found in Ex. 28. 



d v d 2 x 

34. Given e x -f- x = ^ -f- j/, find — ^ and -^ , and show that 



d*y _ (e x+y —i)(x—y) 

~dx^~ (^ 4- 1)3 



XL] EXAMPLES. IOj 



35. Given & + xy — e = o, find ^-£- 



^V _ (2 — >')e y 4- 2X 
fa* ~ y (& 4- xf ' 



d 2 y 
36. Given jfi — $axy -f- ^ 3 = o, find — —^. 



d' l y 2cfixy 



dx* " {f- — ax) z ' 



XII. 

Expressions for the nth Derivative. 

101. When the derivative of a function can be put in a 
form similar to that of the function itself, a general expression 
for the derivative of any order may be written. Thus, be- 
cause De x = e x we have obviously D* 1 ?* = **, where n is any 
positive integer^ So also, since the derivative of sin x, which 
is cos x, may be written sin (x -f- i*r), we have 

Z>* sin (x ~\- a) — sin ( x + # + — ) , 

which includes, as a particular case, 

r^ / , n7t \ 

D n cos # = cos ( x -\ I . 

102. Again, taking the derivative of x n r times in suc- 
cession, we have 

~j^r = n(n— 1) . . .{n—r-\- i)x*~ r , 

which vanishes if n is a positive integer when r > n-\- 1, but 
never vanishes if n is fractional or negative. In particular, 



104 SUCCESSIVE DERIVATIVES. [Art. 102. 

if n is a negative integer, putting n = — *», the equation 
may be written 

-^ = (- i )',«(;« + i) . . . ( M + r - o*-*-^. 

When #z = I, this becomes 

d'' /i \ x r! 

0' 



dx r \x) ' x r+1 ' 

i 
Since Z? log # = — , it follows that 

(r— i>! 

103. The derivative of a function does not necessarily 
bear any resemblance in form to the function itself; but, in 
some cases, a more or less obvious device suffices to reduce it 
to the required form, so as to enable us to express the /zth 
derivative. For example, let 

y = e ax cos (dx) y (i) 

then 

dy = e ax \_a cos (bx) — b sin (bx)\dx. 

Employing an auxiliary constant a determined by 

b = a tan a, (2) 

we have 

dy a 

-j- = ^*-"Tcos (bx) cos a — sin (bx) sin a], 

dx cos a J 

or 

Dy = a sec a e ax cos (bx -j- a). ... (3) 

Therefore the operation of D upon this function is to multi- 
ply by the constant factor a sec a and to add the constant a 



§ XII.] EXPRESSIONS FOR THE nth DERIVATIVE. 105 

to the angle involved. Hence, repeating the operation, we 
have 

D n y = a n sec" ot e ax cos {bx -\- not) ; 

or, since, by equation (2), a sec « = ^/{a 2 -f- b 2 ), 

D n [e ax cos (fo)3 = (a 2 -f ^) 7 *** cos (bx + » tan" 1 -).' (4) 

This formula represents a series of functions of which it will 
be noticed that the original function is the member corre- 
sponding to n = o. 

x 
104. The successive derivatives of y = cot -1 -, though 

bearing no resemblance in form to the original function, yet 
follow a law which is detected by expressing them in terms 
of y. Thus, let 

x 
y = cot -1 — , then x = a cot y ; . . (1) 



differentiating, dx = — a "cosec 2 ;y dy ; whence 

dy sin 2 ^y 

Taking the derivative, we have 

cP;y _ 2 . dy _ 1 



(2) 



dx i a J y dx a* 



sin? cosy^= ^ sin 2y s'm 2 y. . . (3) 
Again, 



d z y 2 . dy 

-^3 = ^ sin y( sin 2 ? cos y + cos 2? sin y) ^, 



106 SUCCESSIVE DERIVATIVES. [Art. IO4. 

and, substituting from equation (2), 

d 3 y 1.2. 

S? = " ~^~ sm 3 ^ sm ^ (4) 

In like manner we obtain 



d*y 1.2.3 

- 4 = -— j-sm^sm^, 



and, in general, 



d n y , . (n— i)\ . 

— =(_i).— -^— sin nysm»y. 

a 
Finally, since from equation (1) sin y 



I d\ n , x , . (n — 1) ! . r . xn 

U) cot ^ =( - i) "^7 ) r+ cot *} (5) 

The nth derivative of tan — is the same expression with 
its sign changed. 

Leibnitz Theorem. 

(05. By means of the following theorem, which is due to 
Leibnitz, the higher derivatives of the product of two func- 
tions is expressed in terms of the successive derivatives of the 
given functions. Let u and v be functions of x ; then 
d(uv) = udv -f- vdu, and using D to denote the derivative with 
respect to x, 

D{uv) = u . Dv -f- Du . v (1) 

Thus the derivative of the product is the sum of two terms of 
which the first is the value it would have if u were constant, 
and the second the value it would have if v were constant. 



§ XII.] LEIBNITZ' THEOREM. IOJ 

Applying this principle in taking the derivative of the 
second member of equation (i), we derive 

D 2 {uv) = u . D 2 v + Du . Dv 

+ Du . Dv + D 2 u . v, 

in which the first line is the result of treating the ^-factor of 
each term as a constant, and the second that of treating the 
^-factor as a constant. Thus we have 

D\uv) — uD 2 v + 2DuDv + D 2 u . v, . . . (2) 

in which the coefficients are those of the expansion of {a -\- b) 2 . 
Again, the application of the same principle to equation (2) 
gives 

D\uv) = uD 3 v + 2DuD 2 v + D 2 uDv 

+ DuD 2 v + 2D 2 uDv + D 3 u . v, 
or 

D\uv) = uD s v + iDuD 2 v + $D 2 uDv + DHi . v, . (3) 

in which the numerical coefficients are those of the expansion 
of {a + b)\ 

In like manner we can derive D 4 (uv), etc. ; and from the 
manner in which the coefficients arise it is evident that they 
will always be identical with those in the successive expan- 
sions of the powers of a -\- b ; that is to say, they are the 
coefficients given in the Binomial Theorem. Hence 

n(n — 1) 
D n {uv) = uD n v + nDuD n ~^v -| D 2 uD n ~ 2 v +. . . 

106. In particular, if we put u = x, Du = 1, the higher 
derivatives of u vanish, so that Leibnitz' Theorem reduces to 
its first two terms. Thus 

D"(xv) — Dx n v + nD n - x v ; . . . . (1) 



108 SUCCESSIVE DERIVATIVES. [Art. Io6. 

hence if we have the expression for the #th derivative of v> 
we can write that of xv. 

For example, given v = log x, using the expression for 
D r log x found in Art. 102, we have 



D n {x log x) = (— i)* -1 
which reduces to 



(n — 1) ! n{n — 2) ! 
x n ~ * x n ~ * 



^ , (n — 2)! 

D n {x log x) = (— \) n 



n-l 



X 

This result is not applicable when n= I, because the symbol 
Z> log x, which then occurs in the application of equation (1), 
cannot be evaluated by putting r = o in the expression for 
D r log ^c. 

In like manner, if u = x 2 , Leibnitz' Theorem reduces to 
its first three terms. 

107. If « = e ax y D"u = a n e ax '; that is to say, as applied to 
this simple function, the operation D has the same effect as 
multiplication by the constant a, and, since at each step the 
operand,, or function operated on, remains of the same form, 
this is true of repeated operations. Now, using this value of 
u in Leibnitz' Theorem, we find 



D n (e ax v) — e ax 



7i(n — I s ) 
D n + naD n ~ l + ^ J - aW n ~ 2 + . . . 



*,(!) 



in which the compound symbol prefixed to v means that the 
results of the operations of the several symbols upon v are to 
be added. 

The result may be written in the form 

D\e ax v) = e ax (D -f a) n v (2) 



§ XI I. J EXAMPLES. IO9 

Here the symbol D -f- a indicates the operation of taking the 
derivative and adding a times the operand itself, and the 
symbolic power indicates the repetition of this operation n 
times. In fact equation (2) may be derived directly from 
the value of the first derivative of e ax v. For 

D{e ax v) — e ax Dv -\- ae ax v 
— e ax (D + a)v. 

The second member is of the same form as the original 
operand, (D -\-d)v taking the place of the function v\ hence, 
repeating the operation, 

D\e ax v) — e ax (D + a)(D + a)v — e ax (D -f afv, 
and so on for higher derivatives. 

Examples XII. 

Find the nth. derivatives of the following functions : 

1 d n y {m -\- n — 1 ) ! 



1. y 



(a — x) m dx n (m — 1) ! (a —x) m+n 



d n y , x (n — 1 ) ! 



1 b 

ox A- n tan x — 

a 



dx n log b (a -f- x) n 

d n y 

3. y = log (1 — mx). — — t — —{n — 1)! m n (1 — mx)~ n . 

d n v n 

4. y = e ax sin bx. —=— = (a 2 -\-b 2 )^e a * sin 

5. y — ^* cosa cos(jt: sin a). D n y = ^* cos « cos ^ s j n a __y na y 

6. y — cos 2 * 1 . D*y = 2 n ~ 1 cos (2^ -f- J»?r). 

_i 1 r 1 1 "I 

a 2 — x 2 2a [_a — x * a -\- xj 

ryn - ni r 1 1 ( — I ) W 

"^ ~~ 2~a\_(a — x) n + 1 ^~(a-\-x) n +i 



no 



SUCCESSIVE DERIVATIVES. [Ex. XII. 



8. y — 



x 



a* — x 4 



Dy 



n ! 



(- 1)" 



9. Prove that 



10. y = .a^ 2 * 



(a — x) n+1 ( a -\-x) n+1 _ 



D m +\x m logx) = 



m 



x 



Dy = 2 H ~\n -f- 2X )e 2x * 

11. y = x 2 e*. D n y = [«(« — 1) -J- 2nx -j- x^e*, 

12. Prove that, when « > 3, 

.6(»-4)l 



Z>* (.r 3 log a:) = (— 1)' 



x 7 



13. y = x sin x. 



d n y . I , 7t\ 1 n\ 

- = x sin ( x + w ~ ) — n cos ( .*• + »— I 



t/jf 



14. If>> = tan _1 jtr, we have 



dy 



(1 +*■)£- = !; 

hence derive the following relation between any three consecutive 
derivatives of tan -1 .* : 

(1 + x 2 )Z) n+1 tan -1 .r -j- 2tixD n tan -1 .* -f- »(» — i)D n ~ x \zxr x x = o. 

15. If_j> = sin -1 .*, prove that 

(1 — x 2 )D 2 y— xDy = o; 
and thence show that the higher derivatives satisfy the relation 
(1 — x 2 )B H+2 y — (a« + i)xB H+l y — n 2 Dy = o. 

16. y = (1 — ^) M ^. 



<£tf 



£ = n ! i (1 — x) n — n 2 x(i — x)"' 1 



— x~\«-* 



x) 



^™ • • * r • 



§ XII.] 



EXAMPLES. 



Ill 



17. If y = x m e ax , prove that 

dy 



dx n 



— &" 



a n x m -f- na n ~ l mx m ~ l 

n(n-i) . 



-\ a n ~ 2 m{m — i )x m ~ 2 -{• . 



I . 2 



3 



and thence show that 



x Ti 



dx r 



e ax x n ) = a m ~ n -r-\e ax x m J. 



x 



18. From the «th derivative of tan -1 — , Art. 104, derive that of 

a 



a 2 -j- x 2 



fin 



a 2 -\- x 2 



a{a 2 -|- x 2 ) 2 *- 



19. If>> = log \Z(a 2 -\- x 2 ), prove that 

2?^ = (— i)*-u J — cos In cot" 1 — ) 



CHAPTER IV. 

Maxima and Minima. 



XIII. 

Characteristics of a Maximum Value. 

108. One of the simplest applications of the Differential 
Calculus is the determination of the greatest and least values 
which a quantity varying continuously under given conditions 
can assume. 

We suppose, at present, that the quantity can be expressed 
as a function of a single independent variable; so that the 
problem is that of determining the greatest or least value of a 
function fix) while x goes through a certain range of values. 

For simplicity we shall always suppose x to increase 
through its range of values. Then, by Art. 39, fix) increases 
so long as the derivative fix) is positive, and decreases so 
long as fix) is negative. Hence, if z is a value of x for which 
fix) is a maximum, f(x) must change sign from -}- to — , when 
x passes through the value z. Except in special cases, to be 
considered hereafter, fix) is a continuous function, and there- 
fore must take the value zero at the instant when it changes 
sign ; hence z must be a value of x which satisfies the equation 

f(x) = o. 

109. For example, let it be required to divide the number 
a into two such parts that the product of the square of one part 
and the cube of the other may have the greatest possible value. 



§ XIII.J 



MAXIMA AND MINIMA. 



113 



Taking the part to be squared for the independent varia- 
ble x y the other part is a — x, and the quantity to be made a 
maximum is 

f(x) = x\a — xf (1) 

The equation f\x) — o becomes in this case 
2x(a — x) 3 — 3^(a — x) 2 = o, 



or 



x(a — x) 2 {2a — $x) = o (2) 



The roots of this equation are x == o, x = a and x = \a. The 
last value only corresponds to a division of a into two parts ; 
it therefore gives the maximum required, and accordingly we 
find, on examining the first member of equation (2), that f\x) 
is positive when x is less than f T a, and negative when x 
exceeds fa. 

The maximum value of f(x) is f{\(i), which, by substitution 
in equation (1), is ■■££%■$<&* 

Maxima and Minima of Continuous Functions. 

.110. When a continuous function changes more than once 
from an increasing to a decreasing function, or vice versa, it is 
regarded as having a maximum or a minimum value whenever 
the change takes place. In other words, a value of a continu- 
ous function which is greater than the neighboring values is 
called a maximum, and one which is less than the neighbor- 
ing values is called a minimum ; 
even though greater values in 
the one case, or less values in 
the other, may exist. 

For example, Fig. 21 is the 
graph of the function in equa- 
tion (1) of the preceding article, 
so that the problem is to find the maximum or minimum 



x<IG. 21. 



114 MAXIMA AND MINIMA. [Art. I IO. 

ordinates of the curve 

y =zx\a —xf y 

which is continuous for all values of x. The point B in the 
diagram corresponds to the maximum value found above. The 
origin and the point A, (fl,o) correspond to the other roots 
of equation (2). Now the first member of this equation 

x(2a — $x)(a — x) 2 , 

or value of f'(x), is negative for negative values of x, and 
changes sign from — to -f- when x increases through zero. 
Accordingly, we have a minimum at the origin as well as a 
maximum at B; although there are values of y to the left 
of greater than the maximum, and values to the right of A 
which are less than the minimum at 0. 

HI. Since fix) = tan the roots of the equation/ 7 ^) = O 
are the abscissae of the points on the curve where the gradient 
is zero, that is, where the tangent is parallel to the axis of x. 
They may be called the critical values of x because they are 
points at which maxima and minima may occur. But any 
one of them may fail to give either a maximum or a minimum. 
For example, in the present case, while x passes through <z, 
f f {x) becomes zero, but does not change sign. In fact f'(po) 
after becoming negative at B remains negative, and the 
original function f(x) continues to be a decreasing function 
for all values to the right of B. 

The various sections of the curve are marked in the dia» 
gram with the sign of the derivative. 

Maxima and Minima of Geometrical Magnitudes. 

(12. When the maximum or minimum value of a geo- 
metrical magnitude restricted by certain conditions is required, 



XIII.] 



GEOMETRICAL MAGNITUDES. 



115 



we seek if possible to obtain an expression for the magnitude 
in terms of a single unknown quantity, that is, to express it 
as a function of one independent variable. 

For example : let it be required to determine the cone of 
greatest convex surface among those which can be inscribed in a 
sphere whose radius is a. 

Any point A of the surface of 
the sphere being taken as the apex 
of the cone, let Fig. 22 represent a 
great circle of the sphere passing 
through the fixed point A. 

If we refer the position of the 
point P in the base of the cone to 
rectangular coordinates, taking the 
centre of the sphere as origin, the 
required cone will evidently be de- FlG 22 

termined when x is determined. We have now to express 
the convex surface 6* in terms of x. 

The expression for the convex surface of a cone gives 

S = ?ty \/\y 2 + (a + xf], . . . . (1) 

in which the unknown quantities x and y are connected by the 
equation of the circle 




x z -f- y 2 = a 2 . 
Substituting the value of y, we have 

S = 7t |/(V — of) \/{2a 2 + 2ax), 
which reduces to 



( 2 ) 



S = 7t ^(2 a) (a -)- x) ^/(a — x). 



(3) 



n6 



MAXIMA AND MINIMA. 



[Art. 112. 



Since the factor n \/{2a) is constant, we are evidently re- 
quired to find the value of x for which the function 

f(x) = (a -[- x) |/(a — x) 

is a maximum. The equation/ 7 ^) = o is, in this case, 

a ~\- x 



\/{a — x) — 



2 \/(a — x) 



= o; 



whence 



x — S^*'* 



Sub- 



The altitude of the required cone is therefore |a, 
stituting the value of x in equation (3), we have 



the maximum value required. 

113. As a further illustration, let it be required to determine 

the greatest cylinder that can be in- 
scribed in a given segment of a parab- 
oloid of revolution. 

Let h denote the altitude, and b 
the radius of the base of the seg- 
ment. The equation of the gener- 
ating parabola is of the form 

y 2 = 4ax. 

Since (h,b) is a point of the curve, 
Fig. 23. we have the condition b 2 = 4ah; 

hence, eliminating 4a, the equation of the curve is 




Y = —x 



(o 



§ XIII.] 



GEO ME TRICA L MA GNIT UD ES. 



117 



The volume V of the cylinder of which the maximum value is 
required is expressed by V = ny\h — x), or, by equation (1), 



Hence we put 



b 2 
V — n—xih — x). 
h 



f(x) — hx — x 2 , 



and the condition f'(x) = o gives 

X — 'o'/Z'. 

Consequently h — x, the altitude of the cylinder, is one-half 
the altitude of the paraboloid. 

114. A problem involving a maximum or minimum some- 
times requires statement in a changed form, 
before the variable can be made a function of B 
a single independent variable. For example : 
required the length of the longest rod which 
can be passed up a chimney of which the width 
is b and the height of the opening above the 
floor is a, the rod being supposed to lie in a 
vertical plane, see Fig. 24. Taking as coordi- Fig. 24. 

nate axes the intersections, OA and OB, of this plane with the 
floor and the vertical back wall of the chimney, it is obvious 
that the rod cannot be greater than any line AB passing 
through the point (a, b) and terminated by the axes. The 
length required is therefore the same as the minimum length 
of the line AB. This length may now be expressed in terms 
of 6, its inclination to the floor. Thus 




AB = /(d) = a cosec 6 + b sec 6. 



Il8 MAXIMA AND MINIMA. [Art. I 1 4. 

Hence, putting 

f{o) = — a cosec 6 cot 6 -f- b sec # tan # = o, 

we obtain 

a 
tan 3 # = T , 

and substituting, we have for the minimum value of AB, or 
maximum length of the rod, 

(a 3 -j- b 3 p. 

Examples XIII. 

1 . Find the sides of the largest rectangle that can be inscribed in 
a semicircle of radius a. The sides are a\/2 and \ai^2. 

2. Determine the maximum right cone inscribed in a given sphere. 

The altitude is four-thirds of the radius of the sphere. 

3. Determine the maximum rectangle inscribed in a given segment 
of a parabola. 

The altitude of the rectangle is two-thirds that of the segment. 

4. Find the maximum cone of given slant height a. 

The radius of the base is \aj\/b. 

5. A boatman 3 miles out at sea wishes to reach in the shortest 
time possible a point on the beach 5 miles from the nearest point of 
the shore; he can pull at the rate of 4 miles an hour, but can walk at 
the rate of 5 miles an hour ; find the point at which he should land. 

Express the whole time in terms of the distance of the required point 
from the nearest point of the shore. 

He should land one mile from the point to be reached. 

6. If a square piece of sheet lead whose side is a have a square cut 
out at each corner, find the side of the latter square in order that the 
remainder may form a vessel of maximum capacity. . 

The side of the square is \a. 



§ XIII.] EXAMPLES. 119 

7. A rectangular court is to be built so as to contain a given area 
c 2 , and a wall already constructed is available for one of the sides ; find 
its dimensions so that the least expense may be incurred. 

The side parallel to the wall is double each of the others. 

8. Determine the maximum cylinder inscribed in a given cone. 

The altitude of the cylinder is one-third that of the cone. 

9. Find the maximum cylinder that can be inscribed in a sphere 
whose radius is a. The altitude is f^^/3. 

10. Through a point whose rectangular coordinates are a and b 
draw a line such that the triangle formed by this line and the coordi- 
nate axes shall have a minimum area. 

The intercepts on the axes are 2a and 2d. 

11. The illumination of a plane surface by a luminous point varies 
inversely as the square of its distance from the point, and directly as 
the cosine of the angle of incidence of the rays; find the height at 
which a bracket-burner must be placed, in order that a point on the 
floor of a room at the horizontal distance a from the burner may re- 
ceive the greatest possible amount of illumination. 

The height is — >• 
6 4/2 

12. A cylinder is inscribed in a cone whose altitude is a, and the 
radius of whose base is b; determine the cylinder so that its total sur- 
face shall be a maximum, and thence show that there will be no maxi- 

mumwhena<2<5. a 2 —2ab 

ihe altitude is — — « 

2 [a — b) 

13. Determine the cone of minimum volume described about a 
given sphere. The height is twice the diameter of the sphere. 

14. A sphere has its centre in the surface of a given sphere whose 
radius is a ; determine its radius in order that the area of the surface 
intercepted by the given sphere may be a maximum. ±a. 

15. Find the point, on the line joining the centres of two spheres 
whose radii are a and b, from which the greatest amount of spherical 

surface is visible. 

— — 
The distance between the centres is divided in the ratio a 2 : b s . 



120 MAXIMA AND MINIMA. [Ex. XIII. 

1 6. Find the minimum isosceles triangle circumscribed about a 
parabolic segment. 

The altitude of the triangle is four-thirds of the altitude of the 
segment. 

17. A tinsmith was ordered to make an open cylindrical vessel of 
given volume, which should be as light as possible ; find the ratio be- 
tween the height and the radius of the base. 

The height should equal the radius of the base. 

18. What should be the ratio between the diameter of the base and 
the height of cylindrical fruit-cans in order that the amount of tin used 
in constructing them may be the least possible ? 

The height should equal the diameter of the base. 

19. Assuming that the expenditure of coal in driving a steamer 
through the water is proportional to the time and to the cube of the 
speed v, find the most economical speed against a current whose speed 
is a. v = J-tf. 

20. In Fig. 24, find the minimum value of the sum of the inter- 
cepts OA and OB. W aJ t Y b ) 2 * 

21. Find the minimum perimeter of the triangle OAB in Fig. 24. 

2\a -j- b -\- 4/(2^)]. 

22. A right cone is cut by a plane parallel to the slant height 
AB. Given that the section is a parabola, and that the area of a pa- 
rabola is f of the circumscribing rectangle ; prove that the area is a 
maximum when the plane bisects the radius OB. 

23. From a point whose abscissa is c, on the axis of the parabola 
y* — \ax, determine the shortest line to the curve. 

The abscissa of the required point on the curve is c — 2a. 

24. Determine the greatest rectangle that can be inscribed in the 



ellipse 



* 2 + ^=i. 



The sides are #4/2 and 3 j/2. 
25. The top of a pedestal which sustains a statue a feet in 
height is b feet above the level of a man's eyes ; find his horizontal 



§ XIII.] EXAMPLES. 121 

distance from the pedestal when the statue subtends the greatest 
angle. 

26. It is required to construct from two circular iron plates of 
radius a a buoy, composed of two equal cones having a common base, 
which shall have the greatest possible volume. 

The radius of the base = \a^d. 
27.' In a given sphere, determine the inscribed cylinder whose en- 
tire surface is a maximum. 
Solution : — > 
Using the notation of Art. 112, we find 

f{x) ■=. a 2 — x 1 -j- 2x4/ (a 2 — x 2 ); 



whence f\ x ) = — 2X ~\~ 2 V(. a<l — x2 ) 



2X* 



^{a 2 — x 2 ) 9 ' 

and/^jr) = o gives 

x^/(a 2 — x 2 ) = a 2 — 2X 2 . . . , ... (1) 
Squaring, we have 

5^ — $a 2 x 2 -\- cfi = o, 

the roots of which are 

^ 2 = a\i ± wi); 

but, since the radical in equation (1) must be positive, we must have 
x 2 < -i<2 2 ; hence the altitude, 2X, of the cylinder is 

a V{ 2 — %vs)- 

28. In a given sphere determine the inscribed cone whose entire 
surface is a maximum. 

The altitude of the cone is — (23 — 4/17). 



122 MAXIMA AND MINIMA. [Art. 1 1 5. 

XIV. 

Discrimination between Maxima and Minima. 

115. We have seen in Arts. 1 10 and in that, in the case 
of a continuous function, the equation f\x) = o may have a 
number of roots, which are the critical values of x to be 
examined for the occurrence of maxima and minima; and 
that, in the graph of the function, these correspond to points 
where the curve is parallel to the axis of x. If one of these 
occurs in a part of the curve which is convex as viewed from 
above, as for example B in Fig. 21, the ordinate/^) is there 
a maximum. If it occurs, like in Fig. 21, in a concave 
part of the curve, f(x) is a minimum. Finally, if it occurs at 
a point of inflection, like A in Fig. 21, there is at the point 
neither a maximum nor a minimum. 

116. It was shown in Art. 98 that, when the value of the 
second derivative f"(x) is negative, the curve y = f(x) is con- 
vex, and when it is positive, the curve is concave. Accord- 
ingly, if f"(x) has a negative value for a critical value of x, 
we have a maximum value of f(x); and ii f"(x) has a. positive 
value, f{x) is a minimum. Thus if f'\x) has a finite value at 
a critical point (that is, a point at which j'(x) = o), a maxi- 
mum or minimum occurs ; but, if f"{x) = o, it is necessary to 
make a further examination to ascertain whether there is or is 
not a point of inflection. 

117. For this purpose, we notice that at any point of in- 
flexion the gradient fix) is either a maximum or a minimum. 
For example, in Fig. 20 f\x) has a minimum value at the 
point of inflexion, while in Fig. 21 its value at A is a maxi- 
mum. It follows that, for a point of inflexion, the derivative 
of f'(x), which isf ;/ (x), must not only vanish but must change 



§ X I V . ] D IS CRIMINA TION BET WEEN THEM. 1 2 3 

sign. Therefore, by Art. 116, if fix) has a finite value, 
there will be a point of inflexion ; and in that case the origi- 
nal function, fix), will have neitJier a maximum nor a mini- 
mum value at the critical point in question. 

118. In the next place, if f'"ix) vanishes at the critical 
point as well as f\x) and f"{x), we examine f v {x). If this 
has a finite value, 'f'(x), of which it is the third derivative, will, 
as shown above, not have a maximum or minimum value ; 
hence there will be no point of inflexion at the critical point, 
and the original function fix) will have a maximum or mini- 
mum value. 

Continuing in this way, we can prove that, whenever the 
first one of the successive derivatives which does not vanish is 
of an even order, there will be a maximum or minimum value; 
but, if it is of an odd order, there will be a point of inflexion, 
and hence no maximum or minimum value. 

In other words,, if all the derivatives of f{x) preceding/" w (x) 
vanish for a certain value of x, while f n (x) has a finite value, 
fix) will have a maximum or minimum value if n is even, but 
not if n is odd. 

119. In the next place, supposing n to be even, we shall 
show that to discriminate between a maximum and a minimum 
we have the same rule, depending on the sign of f n (x), as in 
the case when n = 2, Art. 1 16. 

To prove this, we notice : first, that at a horizontal point 
of inflexion where fix) is a maximum (like A in Fig. 21), 
the function fix) is a decreasing one. Secondly, when at a 
horizontal point fix) is a decreasing function, so that it 
changes its value in the order -\- , o, — , fix) is a maximum. 
It follows that when f n ix) is negative, the preceding functions 
are alternately decreasing ones and maxima. In like manner, 
when f M (x) is positive, the functions are alternately increasing 



J 24 MAXIMA AND MINIMA. [Art. 120. 

functions and minima. Thus when n is even, a positive value 
indicates a minimum and a negative value a maximum. 

120. As an illustration, let us take the function 

f(pc) = e x + e~ x -\- 2 cos x, 
whence 

f\x) = e* — e~ x — 2 sin #. 

In this case, f'{x) = o is a transcendental equation, but it is 
obvious that x = o is a root. We therefore examine the 
values of f"(o) y etc. Differentiating again, 

f"(x) = e x -\-e~ x — 2 cosx, .*. /"(o) = o; 

/"'(a;) = ** — *-* + 2 sin a?, .-. / /7, (o) = o; 

f\x) = e* + e- x + 2 cos *, . •. / IV (o) = 4. 

The fourth derivative is the first one which does not vanish, 
and it has a positive value ; we therefore conclude that f(p) is 
a minimum value of f{x). 

Alternation of Maxima and Minima, 

121. It is obvious that, in the case of a continuous func- 
tion, maxima and minima (when several exist) must occur 
alternately. This fact facilitates the discrimination of these 
values. For example, given 

f(x) = 3# 4 — i6x 3 — 6x? + 12, 

which is continuous for all values of x. Here 

f'{x) = 12X Z — 4&E 2 — I2#. 



§ X I V.] A L TERN A TE O CCURRENCE. 1 2 5 

The roots of f\x) = o are x = o and x = 2 ± 4/5. Again, 

/"(#) = 36X 2 — g6x — 12. 

For the root # = o, we find /"(o) = — 12 ; therefore x = o 
gives a maximum. 

Now, of the other two roots one is positive and the other 
negative, so that zero is the intermediate root. It follows 
that each of these roots gives a minimum of the function. 

122. The same conclusion may be arrived at, in this 
case, as follows: Very large positive values of x make/"(x) 
very large and positive ; hence for the range of values of x 
beyond the greatest critical value, which is x = 2 -\- ^/$, f(x) 
is an increasing function. Therefore this value of x corre- 
sponds to a minimum. ' These conclusions are made clear by 
means of a rough sketch of the graph of the function.* 

123. It is obvious also that, in the case of a continuous 
function, a maximum must be greater than an adjacent mini- 
mum and a minimum less than an adjacent maximum. But 
neither this conclusion nor the alternation of maxima and 
minima can be inferred of the maxima and minima occurring 
in different branches of a discontinuous function. 



* In the same way, we can see that the function in Art. 120 must have at 
least one minimum ; for its values increase indefinitely and are positive both for 
large positive and for large negative values of x. Hence, if zero is the only criti- 
cal value, it must correspond to a minimum. Moreover, that there is no other 
critical value except zero may be shown as follows : f" (x) may be put in the form 

„,, . <? 2 * — 2 cos x . e x -h 1 e™ — 2<?* 4- 1 

f{*) = > ; 

e x e x 

but the last expression (of which the numerator is a perfect square) cannot become 
negative. Therefore f'{x) cannot again become zero. 



126 MAXIMA AND MINIMA. [Art. 1 24. 



Employment of a Substituted Function. 

124. It is often convenient, in determining a maximum or 
minimum, to substitute for the given variable some function 
of it which obviously arrives at its maximum or minimum at 
the same time. For example, to determine the maximum 

value of 

y = |/(6 2 -j- ax) + V (6 2 — ax) . 

It is obvious that the square of a positive quantity will reach 

a maximum simultaneously with the quantity itself. In this 

case 

f = 2b 2 + 2 4/(6 4 - a 2 *?), 

which is obviously a maximum when x = o. We infer that 
y is a maximum when x = o ; the maximum value is there- 
fore 2b. 

125. A decreasing function of a variable (that is, one 
which decreases when the variable increases and increases 
when the variable decreases) will evidently reach a maximum 
when the original variable reaches a minimum, and vice versa* 
Thus, to find the maxima and minima of 

/(*) = 



x 2 — 3#-|- 1 ' 
we may with advantage employ the reciprocal, viz., 



<p(x) = x— 3 +-. 
x 



Taking derivatives, 

4>'(x) = 1 — — , <P"(x) = 2X~ Z . 

XT 

The roots of (p\x) = o are x — ± 1 ', x = 1 makes (f>"(x) 



§ XIV.] USE OF A SUBSTITUTED FUNCTION. 127 

positive ; hence it gives a minimum value of (fi(x), and there- 
fore a maximum value of fix). In like manner, x = — I is 
found to give a maximum value of (p(x), and therefore a mini- 
mum oifix). 

In this example, the maximum value, which is/(i) = — 1, 
is algebraically less than the minimum, which \sf{— 1) = — -J. 
This is accounted for by the fact that the function is discon- 
tinuous; it has an infinite value corresponding to 

*==!— iV5 = -38, 

a value which lies between -j- 1 and — 1 ; so that the maxi- 
mum and minimum points occur in different branches of the 
graph of the function. 

126. In some cases we may use a simplified function in 
place of f ; {x) in discriminating between maxima and minima. 
For example, given the function 

f[x) = —r—, whence f{x) = °*f * ~ . 

log x (log xy 

Since our object is only to ascertain whether f'{x) changes 
sign in the order — , o, -f- or in the order -|-, o, — , we may 
omit the denominator, which is always positive. The numer- 
ator, log x — 1, vanishes when x = e, and its derivative which 
takes the place of f"{x), namely x _1 , is positive when x = e ; 
hence the corresponding value of the function, namely 
f(e) = e, is a minimum. 

Examples XIV. 

1. Show that the function ae kx -\-be~ kx has a minimum value 
equal to 2 \/{ah). 

Determine the maxima and minima of the following functions : 

2. fix) = x x . A min. for x = — . 

e 



128 MAXIMA AND MINIMA [Ex. XIV. 

_ . log X I 

?. fix) = — - — . A max. for x = — . 

X n 6 n 

( d Xy 

a. fix) = - -. A min. for x = \a. 

y x ' a — 2X * 

5. /(.*) = — f- -. A mm. for x = — T \. 

4/(i + 5*) 

6. /"(.*) = 2 cos ^ -f- sin 2 .*. Max. for x = 2nn\ 

min. for x = (2^ -|- i)tt. 

7. /"(.*) = sin ^(1 + cos .*). Max. for x = -|7r; 

min. for x = — ^tt; 
neither for x = n. 

8. f{x) = sec .* -J- log cos 2 .*. Max. for x t = o, and j; = 7T; 

min. for ^ = ± |-7r. 

tan 3 * Min. for x = o, ^n, f ft, and 7r; 

q. fix) = . r , . _ 

7 v tan $x max. for .* = ^7t, \rt, -J7T, etc. 

10. f{x) — e x + e~ x — x 2 . A min. for x = o. 

11. /(.*) = 4** 2 + cos 2Jf — i(* 2 * H" e " 2X )' 

Max. for .* = o. 

12. /(.*) = (3 — x)e 2x — 4^^ — .*. Is there a maximum or a 
minimum corresponding to .* = o ? Neither. 

13. f{x) = .*(.* + tf) 2 C# — «) 3 o 

Min. for x = — <z and .* = -^<z; 
max. for .* = — i<z. 

14. f(x) = sin 2.* — x. Max. for .* = nn + ^-tt; 

min. for :r = «7T — ^7z\ 

15. f(x) = 2X 3 -j- 3^ 2 — $6x-\- 12. Max. for x = — 3; 

min. for x = 2. 

16. /"(.*) = ^ 3 — 3^ 2 — gx -f 5, Max. for x = — 1; 

min. for x = 3. 

17. /"(.*) = 3.x 5 — 1 2$x 3 -j- 2160.*, 

Max. for x = — 4 and x = 3; 
min. for .* = — 3 and .* = 4. 



§ XI V.J EXAMPLES. 129 

18. f{x) = b -j- c(x — a)». Neither a max. nor a min. 

19. /[x) = (^ — i) 4 (.r -j- 2) 3 . Max. for x = — |; 

min. for x = 1. 

20. /(.*) = (.# — 9) 5 (^ — 8) 4 . Max. for * = 8; 

min. for .r = 8|. 
t — r -J- .r 2 

21. A*) = 

22. y(^) = 

23- A x ) = 

24. /(*) = 

25. The lower corner of a leaf of a book is folded over so as just 
to reach the inner edge of the page. Denoting by a the width of the 
page, and by x the part of the lower edge turned over, show that the 
length of the crease is 

x \/x 



I -\- X — X 2 ' 




Mm. for x = -|. 


ax 




Max. for x = 1; 


ax 2 — bx -\- a 


min. for x = — 1. 


x — 1 




Max. for x = 4. 


X 3 — $X 2 -|- 2X 


+ 54* 


x 2 — x -{-. 1 




Max. for x = 0; 


x 2 4- x — 1 " 




min. for x = 2. 



J / = 



i'O* - i«)' 



and thence find j; when_^ is a minimum. . jr = fa\ 

26. Find when the area of the part folded over is a minimum. 



X — ~§Q>, 



XV. 

Implicit Functions. 

127. When y is an implicit function of x, defined by the 

equation 

F(x, y) = 0, ...... (1) 

the first derivative, found as in Art. 87, takes the form 

dy __ u 



dx v 



(2) 



130 



MAXIMA AND MINIMA. 



[Art. 127. 



where u and v are usually functions of x and y. This deriva- 
tive takes the value zero if u = o, provided v does not vanish 
at the same time.* Hence, to find a maximum or minimum 
value of y, we must find the values of x and y which satisfy- 
simultaneously the two equations 

F(x, y) = o and u = o. 

This is the same thing as finding the horizontal points of the 
curve whose rectangular equation is F(x, y) = o. 
128. For example, let us take the equation 



xy 2 — x 2 y = 2 a 3 , 



(1) 



in which a denotes a positive constant. Differentiating, 



v 2 -f- 2xy-- — 2xy — x 2 ~~ = o ; 
ax ax 



therefore, 



dy 

dx 



y{2x — y) 



(2) 



x{2y — x) 

In this example, u = y(2x — y) and v = x{2y — x) ; putting 
u = Oy we obtain 

y = o or y = 2x. 

Substituting y = o in equation (1) gives an infinite value 
of x, showing that the curve has the axis of 
# ociox an asymptote as represented in Fig. 25, 
Next combining y = 2x with equation 
^_ (I), we find 

and 





x 



a 



y = 2a, 



Fig. 25. 



which are the coordinates of the point A in 
the diagram. At this point v does not 



* The case in which u and v vanish simultaneously will be considered in the 
next chapter. See Art. 171 



§ XV.] IMPLICIT FUNCTIONS. 1 31 

vanish, therefore the curve has a horizontal tangent, the ordi- 
nate in this case being a minimum. 

129. When it is necessary to find the value of the second 
derivative at a horizontal point in order to discriminate be- 
tween maxima and minima, the work of finding it, as illus- 
trated in Art. ioo for the general case, can be much shortened. 
Differentiating equation (2), Art. 127, with respect to x, we 
have 

du dv 

v u — 

dy 2 _ dx dx 



dx 2 v 2 ' 

but, since, in the cases now under consideration, u = o, the 
second term in the numerator vanishes. Hence, distinguish- 
ing by brackets the special values which the derivatives take 

when - — = o, we have 
dx 



'du 
d 2 y~\ 






dx_\ 



dx 2 J v 

in which the values of x and y found for the horizontal point 
are to be substituted. For example, in the illustration given 
in Art. 128, we find 



c 



^1 = 2y ; whence *¥\ = ±, 

dx 2 J x(2y—x)' dx 2 ] a ^ a 3 a' 



which, having a positive value, indicates a minimum ordinate 
as in the diagram. 



132 MAXIMA AND MINIMA. [Art. 130, 



Maximum aud Minimum Abscissce. 

130. If, in equation (1), Art. 127, we regard x as an 
implicit function of y, we have, using the same notation, 

dx __ v 
dy u 

Hence the points at which x has a maximum or minimum 
value are found by means of the simultaneous equations 

F(x, y) = o and v = o. 

For instance, in the example of Art. 128, v = o gives x = o 
or x = 2v. Combining these in turn with equation (1), the 
first gives the infinite value of v, indicating the axis of y as 
an asymptote; the second gives x = — 2a, y = — a, the 
coordinates of the point B in Fig. 25, at which the abscissa 
is a maximum. Points of this character, where the tangent 
to the curve is parallel to the axis of y, may be called the 
vertical points of the curve. 

Infinite Values of the Derivative. 

13 1 . When x is regarded as the independent variable and 
y as the function, the vertical points are those at which the 

derivative — takes an infinite value. They are usually points 
dx 

like B in Fig. 25, at which the function y is discontinuous. 

Thus, in the figure, y is a two-valued function for values of x 

less than —2a. The two values become equal when x = — 2a, 

and become imaginary for values greater than — 2a. 

In fact, whenever x regarded as a function of y has a maxi- 



XV.] 



INFINITE VALUES OF THE DERIVA TIVE. 



133 



mum or minimum value, it is evident that the curve lies on 
one side of the tangent in the neighborhood of the point of 
contact ; hence this value of x is, for the inverse function y, the 
limit of a range of values for which that function is continuous. 

When the equation is quadratic for y this gives a convenient 
method of finding maxima or minima values of x. 

132. There are, however, two exceptional cases in which 
a vertical point does not give a limiting value of x, the func- 
tion y remaining continuous when x passes through the value 
in question. In other words, there are cases in which the curve 
crosses the vertical tangent. 

The first case is that in which the 
point of contact is also a point of in- 
flexion. For example, in the case of the 
function 



\/x 



or 



f = x, 



we have — — — 



which is infinite at 




^< 



Fig. 26. 



dx 3 #3 
the origin, but is real and positive on 
both sides of the origin. The curve 

takes the form given in Fig. 26, neither x nor y having a 
maximum or minimum value. 

133. The second case is that in which the curve lies on both 
sides of the tangent at the critical point, but upon the same 
side of the normal. In this case, the curve is said to have 
a cusp. For example, in the case of the function 



__ ^j 



y = x 



or 



y 3 = x 2 , 



dy 



we have-^- = — r , which, as in the preceding case, is infinite 



dx 



3* a 




134 MAXIMA AND MINIMA. [Art. 1 33. 

at the origin and is real on both sides of the origin, that is for 

positive and negative values of x. There 

is, in this case, a minimum value of y be- 

dy 
cause the derivative -7— changes sign from 

ax 

— to -j- as x passes through the value o. 
The curve, which is the semi- cubical parabola, 
takes the form given in Fig. 2 J '. 
FlG 27. This is the exceptional case mentioned 

in Art. 108 in which a maximum or mini- 
mum occurs although the derivative is not zero. The general 
condition for a case of this kind is that when the derivative 
is infinite, the function y shall remain finite and continuous, 
and that the derivative shall change sign. The function y is 
then a maximum or minimum according as the change is from 
-j- to — or from — to -j-, exactly as in the usual case.* 

Functions of Two Variables. 

134. A maximum value of a function f(x, y) of two in- 
dependent variables is defined as a value greater than any 
neighboring value of the function. In other words, f(a, b) is 
a maximum value, if f(x, y) changes from a state of increasing 
to a state of decreasing when x and y pass simultaneously 
through the values x=a and y=b, irrespective of the relative 
value of their rates. In particular, if either of the variables is 

* On the other hand, if in such a case we were considering ^rasa function of y, 

dx 
we should find the derivative -y- = o; but neither x nor its derivative would be a 

dy ' 

continuous function while y passes through the critical value. Thus, in the ex- 
ample, x =y% and — = — -V% both of which become imaginary when y passes 
dy 2 

through the critical value zero. 



§ XV.] FUNCTIONS OF TWO VARIABLES. 135 

assumed constant, the conditions for a maximum must be 
fulfilled when the other varies. 

Similar remarks of course apply to a minimum value. 

135. When a maximum obviously exists, it is easy in this 
way to obtain two relations which must exist between the 
variables, and thus determine the special values of x and y. 
For example, let it be required to divide, a number a into 
three parts such that their product shall be a maximum. 
Here two of the parts are independent variables; but assign- 
ing to one of them any fixed value, it is easily shown that the 
other two parts must be equal if the product is a maximum. 
It follows that all three parts must be equal and therefore 
each part is ia. 

Again, let it be required to inscribe the maximum parallel- 
opiped in a given cone. Here, supposing the height to have 
any fixed value, it is obvious that the base must be the maxi- 
mum rectangle inscribed in a given circle, which is a square. 
Now when the altitude varies, the parallelopiped with a 
square base bears a fixed ratio to the circumscribing cylinder 
which is itself inscribed in the cone. Hence the altitude is 
the same as that of the maximum cylinder inscribed in a cone, 
which is readily found to be one-third the altitude of the cone. 

Critical Points on a Surface, 

136. In general, if we put 

z=f(x y y), .......... (i) 

in order that f(a, b) shall be a maximum or minimum value 

of 0, it is a necessary but not a sufficient condition that the 

dz 
derivative — shall change sign when y = b and x passes 
dx 



I3 6 MAXIMA AND MINIMA. [Art. 1 36. 

dz 
through the value a, and also that — shall change sign when 

x =a and y passes through the value b. We shall consider 
only the usual case in which each of these derivatives takes 
the value zero. Thus the special values of x and y will be 
found among those which simultaneously satisfy the two 
equations 

dz „ 1 dz , x 

_ = and _ = o. ... ( 2 ) 

137. If we regard x, y and z as the three rectangular 
coordinates of a point, and consider the surface represented 
by equation (1), the problem before us becomes that of rinding 
the points on the surface which are at a maximum or mini- 
mum distance from the plane of xy. The points on the 
surface of which the x and y coordinates satisfy equations (2) 
are those at which the tangent plane is parallel to the plane 
of xy. These are the critical points at which maxima or 
minima may occur. But for a maximum it is further neces- 
sary that the surface shall lie below the tangent plane at least 
for a certain region, in the neighborhood of, and completely 
surrounding, the critical point. That is, the surface must be 
convex as viewed from above. 

138. When this is the case, the sections of the surface 
made by planes parallel to that of xz } and passing through or 
near to the critical point, will also be convex as viewed from 
above. The equation of a section of this kind is simply the 
equation of the surface when y is regarded as a constant and 
z as a function of x only. Hence the convexity of these 
curves is equivalent to the condition that z shall fulfil the 
requirements of a maximum when regarded as a function of x. 



§ XV.] CRITICAL POINTS ON A SURFACE. 137 

It is of course also necessary that z should fulfil the condi- 
tions of a maximum when regarded as a function of y. 

In like manner a minimum function of two variables must 
fulfil the conditions for a minimum, both when regarded as a 
function of x and when regarded as a function of y. 

139. When examining a point in a given example, it must 
be remembered however that, although the above conditions 
are necessary, they are not of themselves sufficient to estab- 
lish the existence of a maximum or minimum. 

But if Ave determine the value of x for which 2 is a maxi- 
mum in terms of y, and substitute this in place of x in the 
expression for z, we shall have a function of y which repre- 
sents the greatest of the values of z in each of the several 
sections of the surface made by planes parallel to the plane of 
xz; and if it can then be shown that this function assumes its 
maximum value when y = b, it will have been completely 
demonstrated that this value is a true maximum. So also, 
mutatis mutandis, in the case of a minimum, as illustrated in 
the following example : 

140. Given the function 

z = x z — laxy -f- y 3 , (i) 



whence 



dz 9 dz .-> 



From the simultaneous equations 

3^ — xay =0 and 3^ 3 — $ax = o . . (2) 

we find the critical values [a, a) and (o, o). In order to test 
the values (a, a), we form the second derivative of z as a func- 
tion of x, namely 

d 2 z . 
= ox. 



dx 2 



138 MAXIMA AND MINIMA. [Art. 140. 

This has a positive value for every point near the critical 
point (a, a), indicating a minimum. Now, from the first of 
equations (2), the value of x which makes z a minimum is 
x = \/(ay). Substituting this in equation (1), we have for the 
minima values of z corresponding to different values of y, 

z = y s — 2asy*; (3) 



whence 



dz 31 d 2 z s- o 3-4 



dy~" *"'' df 

The first derivative is zero when y = aas before found 

'and the second derivative has a positive value. This indicates 

that the value of z corresponding to y — a is a minimum 

among the minima represented by equation (3), therefore it 

is a true minimum. 

Considering next the critical values (o, o), we find that the 
second derivative with respect to x vanishes for x = o, and 
since the third derivative has the finite value 6, there is, 
according to Art. 117, neither a maximum nor a minimum at 
the origin. 

Examples XV. 

1. Given 257* — 6xy-\- x 2 — 9 = o, determine the maxima and 
minima of> Min. for x =5 — £ ; max. for x — f . 

2. Given x*- -f- 2ax 2 y — ay 3 = o, find the maxima and minima 
f „ Min. for x = ± a. 

3. Given y 2 — x 2 y -j- x — x % =1 o, prove that .# — — 1 gives a 
maximum value ofjy. 

4. Given 3 a 2 y 2 -j- .*y 3 + ^ax % = o. Show that when x = |<z, 
^/ has a maximum value, namely — 3a. 

5. Given j' 3 -j- x 2, — $axy = o, find maxima and minima of y. 

Max. for ^ = a \/2 ; 
min. for x = o. 



§ XV.] EXAMPLES. 139 

6. Given x 2 {y 2 -j- 1) = ^H(xy -\- h), find the maximum value of 
x by the method mentioned in Art. 131. x = 2\/{H 2 -f- Hh). 

Find maxima and minima of the following functions : 

7. f{x) = {pc* — b*)*. Min. for x = o. 

8. /(.#)= (^ - b 2 )i. Max. for * = o • 

min. for x = ± 3. 

9. /(.*) = (x 2 + 3-* + 2 ) f + **• 

f\x) = 00 gives min. corresponding to x= — 2, x== — 1 and x=o. 
f(pc) = o gives two intermediate maxima. 

10. f{x) — [x 2 -f- 2x)% — (x -f- 3)3. Max. for .r = £( — 3 ±4/ 17); 

min. for x = o and Jf = — 2. 

11. /"(■*) = (x — a)* (x— 5)o -j_ c. Max. for .r = — » ; 

min. for x = a and x = b. 
_ (x — a) (x — b) 2ab 

12. XW — 2 ' Mm * for A " — 



a 4- 3' 



13. f{x) = (x — #)f (.r — 3)3. 



Solutions for x = a and x = \{2.b -\- a) • if 3 > 0, the former 
gives a max. and the latter a min. 

14. A number is to be divided into three parts, such that the 
product of the mth power of the first, the »th power of the second, 
and the />th power of the third shall be a maximum. Show that the 
parts will be in the ratios m : n : p. 

15. Show that the polygon of given perimeter and number of sides 
has a maximum area when equilateral and equiangular. 

16. Show that the function 

x 2 — $xy -\- 2y 2 + 4X — 2_y -\- 3 

has a minimum value corresponding to x = 10, y = 8. 

17. Show that the function 

a 2 -f- b 2 — x 2 —y 2 —2b\/(a 2 —y 2 ) 



I40 MAXIMA AND MINIMA. [Ex. XV. 

has a maximum when # > b, but neither max. nor min. when a < h 
and when a = b, 

1 8. Determine whether 



z = V'a 2 —y 1 + ^ 
has a. maximum or minimum value. 



CHAPTER V. 

Evaluation of Indeterminate Forms. 



XVI. 

Indeterminate or Illusory Forms. 

141- A quantity given in the form of a fraction is inde- 
terminate in value if both numerator and denominator have 
the value zero and admit of no other value. 

Suppose now that the terms of the fraction are continu- 
ous functions of x which becomo zero for a particular value, 
say a, of x (but are in general not equal to zero) ; then the 
fraction is itself a continuous function of x, and is said to 

o 
take the indeterminate form — , when x = a. Such a fraction 

o 

has definite values, which for all values of x except a can be 

found by division. As x approaches indefinitely near to a, 

these values approach indefinitely near to a certain value 

which is often called the limiting value of the fraction ; and, 

in order to make the fraction a continuous function of x when 

x passes through a, it is necessary to regard this limiting value 

as the value of the fraction corresponding to x = a. 

142. For example, it is shown in Trigonometry that, if 

jx stands for the arcual measure of an angle, each of the ratios 



142 EVALUATION OF INDETERMINATE FORMS. [Art. I42. 

sin x , tan x . . , _ . , 

and approaches indefinitely near to unity, when 

x x 

x is made indefinitely small. We therefore assign unity as 

the value of either of these ratios when x = o. In this way 

only can we regard them as continuous functions of x when 

x passes through the value zero. These and similar results 

are expressed by using the value of the independent variable 

as a suffix: thus 



sin x~] , tan x~ 
and 



n x~~\ 

00 _Jo 



X 



= I. 



o 

143. The form — may be regarded as the standard inde- 
terminate form, but the term is also applied to functions 

00 
which, on direct substitutions, take one of the form — , 

00 X o, 00 — 00 , and to certain forms whose logarithms take 
the form 00 X o. 

The term illusory is also applied to each of these forms 
because their evaluation requires some process other than the 
operation directly implied in the form of the function. 

144. In some cases, a function in the standard inde- 
terminate form can be evaluated by making an algebraic 
transformation which permits the cancelling of the factor 
which causes the terms to vanish for the given value of x. 
For example, the function 

a — i/(a 2 — bx) 
x 

o 
takes the form — when x = o. Multiplying both terms by 

the complementary surd a -f- V( a2 ~ bx), 



XVI.] EVALUATION BY DIFFERENTIATION. 



H3 



we obtain 



bx 



x[a-\- \/{a 2 — bx] a -\- tf(dr — bx)' 

The last form is not illusory for the given value of x, since 
the factor which becomes zero has been removed from both 
terms of the fraction. The value of the fraction for x = o is 
therefore 

a - 4/(V - b 2 )l b 



x 



2a 



Evaluation by Differentiation. 

145. The principles of differentiation afford us a general 

method by which we can derive from a function, which takes 

the standard indeterminate form when x = a, another function 

which, although not generally equal to the given one, has the 

same value when x = a. 

v 
For this purpose, let - represent a fraction in which both 

u 

u and v are functions of x, which vanish when x = a ; in other 
words, for this value of x t we have u = o and v = O. 

Let P be a moving point of which the abscissa and ordinate 
referred to rectangular axes are simul- 
taneous values of u and v (x not being 
represented in the figure); then, de- 
noting the angle POU by d, and the 
inclination of the motion of P to the 
r axis of u by 0, we have 



v 
tan 6 — - , 
u 



dv 
and tan </> = -=-. 

du 




Fig. 28. 
At the instant when x passes through the value a, u and v 



144 EVALUATION OF INDETERMINATE FORMS. [Art. 1 45. 

being zero by the hypothesis, P passes through the origin ; 
the corresponding value of 6 is evidently determined by the 
direction in which P is moving at that instant, and is there- 
fore equal to the value of at that point. In other words, 
the limiting direction of the secant OP is that of the tangent 
at 0. 

Hence the values of tan 6 and tan corresponding to 
x = a are equal, or 



v 
u 



(kri 



v 
therefore, to determine the value of — for x = a, we substi- 

u 

dv 
tute for it the function -3-, which has the same value as the 

du 

given function when x = a, although differing from it when 

x has any other value. 

146. This result may also be expressed in the following 

manner: Let f(x) and (p(x) be two functions, such that 

/(a) = o and 0(a) = o ; then 

Aa)^f(a) 

0(o) ~<P'{a) w 

log X 

As an illustration, let us take . When x == I, this 

x— 1 

o 
function takes the form — ; by the above process, we have 

log af] x' 1 "1 
the required value. 



§ XVI.] E VAL UA TION B Y DIFFERENTIA TION. 



145 



dv fipc) 
14.7. Since the substituted function -3- or , . . . frequently 

du (x) 

takes the indeterminate form, several repetitions of the pro- 
cess are sometimes requisite before the value of the function 
can be ascertained. 

1 — cos 6 o 

For example, the function 7^ takes the form — 

when 6 = 0; employing the process for evaluating, we have 



cos 0' 



& 



sin 6 
~20~J 



which is likewise indeterminate ; but, by repeating the pro« 
cess, we obtain 



1 — cos 6' 

¥ 2 . 



sin d 



cos d~] _ I 



148. If the given function, or any of the substituted 
functions, contains a factor which does not take the indeter- 
minate form, this factor may be evaluated at once, as in the 
following example. 
The function 

(1 — x)e x — 1 
tan 2 x 



is indeterminate for x = o. By employing the usual process 
once, we obtain 



(1 — x)e? 



tan 2 # 



-]; 



— xe* 



2 sec 2 # tan x 



which is likewise indeterminate; but, before repeating the 



I46 EVALUATION OF INDETERMINATE FORMS. [Art. 1 48. 

process, we may evaluate the factor 5— . The value 

r J 2 sec^J 

of this factor is — ; hence we write 



( 1 — x) e* 



tan' 



g* — l~l xe* ~| 1 x I 

x J 2 sec 2 # tan x_j 2 tan x 2' 



149. In like manner, a factor which takes the indetermi- 
nate form but has a finite value may be evaluated sepa- 
rately. Thus if the given function be 



(e* — 1 ) tan 2 # 



x* 



i- 



tan x~ 1 
knowing that = 1, we may write it in the form 

/tan x\ 2 e^ — i~| 
\ x ) ' x j " 

The second factor is found on evaluation to have the value 
unity ; hence the value of the given function is unity. 

150. Again, if the given function can be separated into 
parts which have finite values, the parts may be evaluated 
separately. As an illustration, we take the expression 



( e * __ e -*y _ 2x\e x + e~ x ) 



U = —Za 

•* — 10 



1- 



The process of separation into parts does not immediately 

e* -f- e~ x 
apply, since the second term reduces to —2 — -^ — , which is 



XVI.] FUNCTIONS WHICH VANISH WITH X. 



H7 



infinite when#= o. But, after making one application of the 
differential process, we have 



u n = 



2(e* — e~ x ){e + e~ x ) — 4x(e + e~ x ) — 2x\e x — er x ) 



Here the last term reduces to — 



4x° 
e?—e~ x ^ 



i 



2X 



e x -{-e x 



1 



I. 



The rest of the expression contains the factor e* -f- e~ x , which 
has the finite value 2 when x = o. Hence the whole expres- 
sion reduces to 



u n = 



e* — e x — 2x 
x s 






and, evaluating, 



Wn 



3^ 



6x J 



Functions which vanish with x. 

151. A function of x which vanishes when x = o will gen- 
erally be found to have a finite ratio to some integral power 
of x. Thus, itf(x) is a function such that f(o) = o, we have 

-^- = f f (o), giving a finite value for the ratio oif(x) to the 

X Jo 

first power of x, unless f '(d) is either infinite or zero. But, 
if f(6) = o, we have 



x 2 



/(*)! _/"(o) 



2X 



! 1= 



giving a finite ratio to x 2 , unless the second derivative is in- 



1 48 EVALUA riON OF INDE TERM1NA TE FORMS. [Art. I 5 I . 

finite or zero. In like manner, if all the derivatives preced- 
ing/*^) vanish when x = o, while that derivative has a finite 
value, we shall find 



f(x) 



f»(o) 



x Jo n \ 

For example, each of the functions sin x and tan x van- 
ishes with a finite ratio (namely unity) to the first power of x, 
because its first derivative has a finite value when x = o. 
Again, I — cos x vanishes with a finite ratio to x\ because its 
first derivative, sin x, vanishes when x = o, but its second 
derivative does not vanish. 

152. The vanishing quantities whose limiting ratios are 
considered above are often called infinitesimals, an infinitesi- 
mal being defined as a variable whose limit is zero. Then, tak- 
ing x as the standard infinitesimal, x 2 is called an infinitesimal 
of the second order. Thus, we have seen above that, when x is 
infinitesimal, sin x and tan x are infinitesimals of the first order, 
and I — cos x is one of the second order; in fact its ratio to 
x 2 is the finite quantity \. Again, tan x — sin x is, under the 
same circumstances, an infinitesimal of the third order ; for 
it may be written in the form tan#(i — cosx), which is the 
product of two infinitesimals of the first and second orders 
respectively. Accordingly, 



tanx — sin x 


1 


X 3 _j 


2 


Examples XVI. 


Evaluate the following functions: 


x z — 5Jt: 2 +7^ — 3 


when 



I 

= 3* — « 

x° — x* — $x — 3 ' 4 



XVI.] EXAMPLES. 149 



x 4 — 8x 3 -f- 22^ — 24JC -4- 9 1 

— -. = ^— ; ; — s when x = 3. — , 

je 4 — 4^ — 2JT 5 -j- I 2.* 4~ 9 4 



i/j; — 4/0 -r */(x — a ) 1 

3- VFZ2 — ^2\ > x = a. 



(i-cosf), 



mx ma 



^(x 2 — a 2 ) ' V( 2a )' 



X k/ixX — 2X^) — XT 8l 

4. 1 ' x = I . . 

1 — j\r 20 

(a 2 4- tfje 4- x 2 )^ — (a 2 — ax -\- x 2 )^ 

5 - ( Z. * ) a ' x=0 - -V'- 

(a -f #)* — (a — xy 

p x p — * 

6. — : -, x—o. 2. 

log (14-*) 

a n — x n 

7. ■ : , x = a. na n . 

log a — log x 

xe 2x — e 2x — x -J- 1 

8- 5T ' 

e~* — 1 

sin jf — cos .r 

9. -, 

Sill 2JC — COS 2X — I 



X = O. — I. 



log -T 

10. ——2 -, X= I. o. 

4/(1 -X) . 

(2* — 3-* a 

11. , ^=0. lOff -r. 

a: ° £ 

4/(1 4- ^ 2 )(i — •*) 4/2 

12. -^ — ', J? = 1. -*— . 

1 — x n n 

a 2 — x 2 ( x\ 1 



I*. 5 — I— COS — ), X = o. 



14. , a: = a. me ma . 

x — a 

a sin x _ a 

15. -; = > x = jt7t. a losr a. 

log sin # * & 

T — COS J*" I 

10. ■ r-, x=o. — . 

X log (I 4- X) 2 

a/x tan x 
17. -|, x = o. 1. 

0* — i) 5 

_, sin x — x cos .*■ 

l8. ; , X=0. 2. 

x — sin x 



150 EVALUATION OF INDETERMINATE FORMS. [_Ex. XVI. 



e x — e~ x — 2.x 

io. , 

' x — tan .*■ 



when x = o. — i. 



(.# — 2)^ 4- •* -4- 2 1 

20. 7^ rx- 1 , .* = O. — . 

x(e* — i) 2 6 

.r — Jt: 

21. — = , X=I. —2. 

1 — x -j- log Jt 

tan 2 Jt: — sin 2 * 

22. -j ., * = o. I. 

(x — i) 2 4- sin 3 (x 2 — 1)$ 

23.-^ '— 1— ^— -! S *=I. ^2. 



(jf -)- l)(jf — i)' 

1 — je 4- log ^r 
1 — |/(2.v — j*: 2 ) 3 



24- Z - „.'. VA > * =1 ' - 1 ' 



sin „v — lo^ (e x cos jt) • 1 

25. ^ S x=o. 

D x z 2 

, J-7T — tan - 1 x t 

26. 5- -TTT^T^, #=1. 



JC" — ^(l***)' 2(l — »)' 

tan (a-\- x) — tan (<z — x) 

27. 77 — — ( \-. '—, x = o. ( 1 4- a 2 ) sec 2 a. 

1 tan -1 + .*) — tan _1 (a — x)' v ~ ' 

n x sin .# — -J-7T 

28. — . ^ = *7r. 



cos ^: 



1. 



ex — e 



sin a; 



2D. ; , X = O. I. 

jv — sin x 



a \ogx _ x 



30. . log X ' 

cos~ *(i — jr) 
|/(2Jtr — x 2 ) 



x== 1. log tf — I. 



31. —777^ T^T* *=°. I- 



Jt: 2 — a a/ ax 

32. — - — r , x = a, $a. 

\/{ax) —a 

tan nx — n tan x 

33- — : = > ^ = 0. 2. 

« sm x — sin #Jtr 

4/2 — cos x — sin x 

34- w ..■;„ „.». > x=\rt. —iV 2 - 



35- 



log sin 2X 

x -\- x 3 — (2» 4- i)^ 2w + 1 4-( 2W — i)^ 2M+3 



(I - .* 2 ) 2 

X = I. tt 2 . 



00 



§ XVII.] THE FORM—- I^I 

m x sin nx — n x sin mx 

36. , when x = o. 1. 

tan nx — tan mx 

tan -nx — tan mx 1 

37- • ■ , o -2-A, ^ = 0. 



sin (n 2 x — m 2 x)' m -\-ri 

tan nx — tan zw.* sec 2 «j*r 

38. - — — 5— r, m = n. 

J sin (»^ — #r.xr) 



2« 



m x sin #.# — «* sin mx 

7.Q. , m = n. 

tan nx — tan mx 



n*~ \n cos xn — sin »Jtr) cos 2 nx. 



XVII. 

1 he rorm — . 

00 

153. \i f{po) and 0(#) are continuous functions of x both 

of which increase without limit when x approaches a given value 

fix) 
a, the function •£* ta kes the form — when x = a. This, 

<?{%) 00 

like — is an indeterminate form, because its value cannot be ob- 
o 

tained directly by division ; but, since the fraction is itself a 

continuous function, its value is not indeterminate, being in 

fact the limit to which the ordinary values of the function 

approach as x approaches a. 

154. It is sometimes possible to ascertain this limit by 
removing from both terms of the fraction by division a factor 
which renders them infinite. For example, the function 

x — sin x 
x-\- cos# 



152 EVALUA TION OF INDETERMINA TE FORMS. [Art. 1 54. 



takes the form — when x = 00 . But, dividing both terms 
00 

by x, we have 



x — sin x 
x-\- cosx 



sin x 

X 



cosx 

X 



~. .,, , sin x . cosx 
Since neither sinx nor cosx can exceed unity, and- 

x x 

vanish when #=00 ; therefore 



x 



sin x 



] - ■■ 



x -f- COS X, 

155. We can, in the same way, evaluate any algebraic 
fraction when x is infinite. For example, dividing both terms 
by x 5 , and then making x infinite, we have 

4x 5 + 5^+3*+ IO ~ 



8x 5 



3* 2 + 3 2 - 



It is obvious that, in this process, all the terms except those of 
the highest degree disappear from the result, which is there- 
fore the ratio of the terms of highest degree. The value is 
finite only in case the numerator and denominator are of the 
same degree. 

156. A fraction which takes the form 00/00 can be so 
transformed as to take the standard form 0/0, and then treated 
by the differential process. For this purpose, the reciprocal of 
the denominator is placed in the numerator, and that of the 
numerator in the denominator. For example, 



sec3#" 
secx 



00 
00 



cosx 



o 
o 



— sin x 



_j in cos3xJ* ff O — 3 sin 3^^ 3 

Since the formula for the derivative of a reciprocal involves 



§ XVII.] 



DIFFERENTIAL FORMULA FOR -■ 

oo 



153 



the square of the original function, this method fails to sim- 
plify the function * unless some transformation takes place, 



such as that in the example above, where 
the simpler function cos x. 



sec x 



is replaced by 



Differential Formula for the Form — ■. 

v 
(57. Let —represent a function in which both u and v are 
u 

functions of x which increase without limit as x approaches 

the value a. Let P be a moving point of which the abscissa 

and ordinate referred to rectangular axes are simultaneous 

values of u and v : then, denoting the angle POV by 0, we have, 

as in Art. 145, 

v 



tan 6 — 



u 



dv 
tan = — . 
du 



In this case, as x assumes the value a, u and v by hypothesis 

become simultaneously infinite, and 

therefore the point P recedes to an 

infinite distance from the origin. Let 

OQ be the limiting direction of OP. 

Then, assuming that the motion of P 

tends to have a fixed direction (which 

will generally be the case) , f it is obvious 

that the limiting direction of OP will 

be that of this motion ; in other words, 




Fig. 29. 



* Thus 



d 



-/'{*) 



The method may, however, be used to prove the 



dx f{x) ~ [f(x)f 

theorem geometrically demonstrated below. 

-j- The example given in Art. 154 furnishes a case in which the motion of J 3 does 

dz> 1 cos x 

not tend to a fixed direction. For in that case tan d> = -7— = : . Both 

du 1 — sin x 



1 54 EVALUA TION OF INDETERMINA TE FORMS. [Art. 1 57. 

we shall have, when x = a, tan 6 = tan 0. Thus 



v~l _ dv 
uj x=a du 



J. 



when _ takes the form — ,* as well as when it takes the 

u °° 

form — . 
o 

This result, like that of Art. 145, may be otherwise ex- 
pressed thus: if 0(a) = 00 and 0(a) = 00, then 

/(a) = f(a) 
0(a) 0'(a)' 

158. Since a variable cannot become infinite in a finite in- 
terval of time while its rate is finite, a function of x cannot 
become infinite, for a finite value of x, unless its derivative with 
respect to x is infinite. It follows that, in the application of 
this formula to a case in which a is finite, the substituted func- 

tion-4r7-x will also take the form — . Hence, in order to effect 

(X) CO 

the evaluation, we must be able at some point in the process 
to make a transformation similar to that made in Art. 156. 
Thus, in the example 

log sin 2x~ 



log sin x _ 



00 
00 



numerator and denominator vary periodically between o and 2, thus tan is a 
periodic function. The curve described by F, in this case, is a prolate cycloid 
with its axis inclined at an angle of 45 to the axis of u. 

* When, as in the diagram, the limiting ratio is a finite quantity, the curve will 
usually have an asymptote AB whose inclination is the limiting value of 0; but this 
is not always the case, for the distance of P from OQ may increase without limit, in 
which case the curve is said to have a parabolic branch in the direction of OQ, 



§ XVII.] THE FORM O X 00. 155 

by using the above formula we obtain 

log sin 2x~] 2 cot 2x~ 



1= 

— I r> 



log sin x J „ ~ cot # 

o — • o — ' o 

which takes the form oo /oo ; but the last expression is equivalent 

to 2— , and is therefore easily shown to have the 

sin2#cos#J 

value unity. 

The Form o X oo . 

159. A function which takes this form may, by introduc- 
ing the reciprocal of one of the factors, be so transformed as 

O oo 
to take either of the forms — or — as may be found most con- 

o oo J 

venient. For example, let us take the function 

which, supposing n positive, assumes the above form when 

X =oo 

Thus, 



00 

x =oo . In this case it is necessary to reduce to the form — . 



e 

x -n e x _ 



nx J n{n — i )x* 



By continuing this process, we finally obtain a fraction whose 
denominator is finite while its numerator is still infinite. Hence 
we have, for all finite values of n, 

x~ n e] — oo . 

[60. When x is infinite, and n > I , x H may be called an 
infinite of the nth order, with respect to x as the standard in- 



156 EVALUATION OF INDETERMINATE FORMS. [Art. 160. 

finite. Thus an infinite of higher order has an infinite ratio to 
one of lower order. The result found in Art. 159 shows that, 
when x is infinite, e* is an infinite of order higher than that 
indicated by any finite number; that is to say, it bears an 
infinite ratio to every power of x. 

In like manner, log x, when x is infinite, is an infinite of a 
lower order than that indicated by any positive value of n. 
For 



log a?" 




~ nx H J ° ' 



thus logx bears a zero ratio to any positive power of x, when 
x is infinite. 



Limiting Values of Discontinuous Functions. 

161. The cases which we have hitherto met in which 
the continuity of a function f{x) is broken belong to one 
or the other of two kinds. In the first kind, the func- 
tion becomes imaginary when x passes a particular value, 
say x = a. That is to say, f(x) is imaginary for values on 
one side of a and real for values on the other side. The 
limiting value, f(a), of the function is usually finite, being the 
value common to two values of a multiple-valued function 
which become equal. This is illustrated by Fig. 15, p. 70, 
the graph of the functions sin -1 #; also, in the case of an im- 
plicit function, by Fig. 25, p. 130. 

In cases of the second kind, the function increases with- 
out limit when x approaches a, and it is usually found, as 
illustrated by the graphs given in Fig. 3, p. 6, and Fig. 13, 
p. 64, to have positive values on one side of x = a and nega- 



XVII.] 



DISCONTINUOUS FUNCTIONS. 



157 



tive values on the other. In such a case, there is no proper 
value of f(a) ; but we may say that the limiting value of f{x), 
when x approaches a from one side is -f- 00, and when x ap- 
proaches a from the other side it is — 00. Thus the recip- 
rocal — is said to pass through infinity and change sign when 
x 

x passes through the value zero. 

162. Consider now a function of a function which becomes 
infinite when x = a. The limiting values to which the com- 
plex function approaches when x approaches a from one side 
or the other may be finite and equal. For example, tan _1 s 
admits of the limiting value \n whether z = 00 or z — — 00 ; 

accordingly tan -1 — has the finite value \rt when x passes 

x 

through zero. There is then no break of continuity. 

On the other hand, e* has the value 00 when z = -f- 00, 
and the value o when z = — 00 ; ac- 

cordingly e*~ has the limiting value 00 
when x approaches zero from the posi- 
tive side, and the limiting value zero 
when x approaches zero from the nega- 
tive side. Thus the function presents 

a third sort of discontinuity.* Its 

1 
graph, that is the curve y = ex, which 

is given in Fig. 30, is said to have a 

stop-point at the origin. 



Fig. 30. 



* The function e x may be said to have a break of continuity when x = oo, 
while tan - 1 x has not. Accordingly the graph of e x , Fig. 10, p. 57, approaches the 
asymptote atone end only, while that of tan - l x, Fig. 17, p. 71, has at each end 
a branch approaching the asymptote. 



158 EVALUA TION OF INDE TERMINA TE FORMS. [Art. 1 63 . 

The Form oo — oo. 

163. It is obvious that the difference between two func- 
tions, each of which increases without limit as x approaches 
a, may approach a finite limit * : hence 00 — 00 is an illusory- 
form. A function which takes this form can be so trans- 
formed as to take the standard form. For example, each 
term of the function • 

1 _ og ( + *) 
x(i -f- x) X* 

takes an infinite value when x = o. But, reducing to a com- 
mon denominator, we have 



log (1 + x) m 



x — (1 -f- x) log (1 -f- x)n o 



x(i -{- x) x 2 J ^(i-f-tf) 

Hence 



Jo o 



r 1 log (1 -f- x)~\ _ x — (1 + x) log (1 -J- x)~] 

[_x(i + x) X 2 J ~ X 2 J 

_ 1 - log (I + x) - i n _ __ l_ 
2X J 2* 

Examples XVII. 

Evaluate the following functions : 

log sec x ■ 

1. =— , when ^ = iar. — 1. 

log sec $x a 

a x 

2. 7 -, X = 00 . AS. 

cosec (mcr x ) 

* This can, of course, happen only when the ratio of the two quantities has 
unity for its limit when they become infinite. 



§ XVII.] EXAMPLES. 159 

3. — - — , («>o), when x = 00. o. 

tan x 

4. . — -, x = \n. 00. 

sec (Irtx) 

5. . t-2 J —, x = 1. 00. 

log (1 -.*)' 

, log COS (^7tx) 

6. , , \ > x = 1. 1. 
log (1 — .*) 

tan ^: 

7. , x = ±n. 3. 

' tan $x ' 2 ° 

log (1 + x) _ 

o. ■ , X — 00 . o. 

X 

a ) 

9. \a — i/x, 

x 2 — a 2 nx 

10. 5 — tan — , 

a 1 2a 

11. x m (log x) M , (m and n being positive,) 

1 

12. e* sin — , 

x 

13. e~l{i —logx), 

TtX . I 

14. sec — . log—, 

2 x 

log tan «jc 

J £? 2 

log tan x ' 

x 
log cot — 

, 2 

10. -. f X = o. o. 

cot x -j- log ^r 

17. sec .%■ (.r sin jr — \7t) 9 

_ , / .r\ 7T^r 

18. log 2 ) tan — , 

\ a J 2a 

19. (1 — x) tan (J7rj;), 

20. log (x — 0) tan (x — #), 



^r = 00 . 


log tf, 


.# = a. 


_4 

71 


x = 0. 


O 


j? = 00 . 


00. 


AT = O. 


0. 


X = I. 


2 
7T* 



^ = o. 



.# 


— 


fr. 


— I. 

2 


.*: 




tf. 


7T 
2 


^ 


— 


I. 


7r' 


AT 


~ 


£. 


O, 



l6o EVALUATION- OF INDETERMINATE FORMS. [Ex. XVII. 

"■ ^ ~ *f COt { 7^S) } ' When X = a - 

Denoting the arc by 6, and multiplying by — -— {whose value, 

6 

when x = a, is unit}'), we obtain — (a -f- x) \ ■— . 

when x = o. oo and o. 



22. 


1 






2 3- 


sec M jr 






,, raw « ' 




24. 


X — X 2 


\og(i 


+ ^)> 


Put 


I 

X = — . 








25- 


2 


I 




J*: 2 — 1 


JT — 


7' 


26. 


cosec .r 


sin 


- 1 * 


jt 


x 2 sin .r ' 


27. 


2 

cot 

X 


"2^, 




28. 


x tan ^ 


— \n 


sec ^r, 


29. 


„r 


1 




x — 1 


log :*• ' 


3°- 


X 


■ cot 2 x 




sin 3 jv 


» 


3 1 - 


1 

2X 2 2 


7t 




\x tan 


Tr.*' 


3 2 - 


I 


1 




4Jt: 2x (e nx 


+ !)' 




7TX — I 




7t 



x = £77-. o. 



.# = — 00 



# = I. 



.# = o. 



Jt* = I. 



X = o. 



.*• = o. 



.r = o. 



1 
2 
1 
6' 



^ = o. o. 



X = \7Z. — I. 



I 
2' 

I 

6' 

7t 2 

T # 

7C 
8* 

"2^ r ^(e 2 ^ — i)' x — o. 6> 

34. Prove that, when /(a) = 1 and (p(a) = 1, 

log 0(a) 0'(*V 



§ XVIII.] LOGARITHMS OF THE FORM 00 X O. l6l 

35. Prove that, when/(V) = o and <p{a) = o, 

log m 

log <p(a) 

provided that ,,, '- is neither infinite nor zero. 
0(«) 



XVIII. 

Functions whose Logarithms Take the Form ooxo. 

164, If u and v are functions of x there are three cases 
in which the logarithm of the function u v takes an illusory 
form. We have, in fact, 

log u v = v log u. 

This is indeterminate when, for a particular value of x, 
v — 00 and log u = o, in which case the given function 
takes the form I 00 : it is also indeterminate when v = o and 
log u = ± 00 , in which case w w takes one of the forms oo° or 
o°. In each of the three cases, the function is evaluated by 
first evaluating its Napierian logarithm. 

165. F° r example, the function 



{■+=)' • 



y — 

X 



takes the form I 00 when x increases without limit. From 
equation (1) we have 

/ _ l0 ^( I +^) 

log^ = *log [1 +-) - — 



1 

X 



l62 EVALUATION OF INDETERMINATE FORMS. [Art. 1 65. 

O 

the last expression taking the form — when #=00. In 

evaluating it, we may conveniently put z for — , then 

x 



log (1 + 2) 1 1 1 



log(i + z) 

^g y\„ = 

Hence 



*L = (1 + £)*] = e - 



The Napierian base is in fact often denned as the limiting 
value of the function in equation (1) when x increases with- 
out limit. When the graph of the function y is drawn it 
is the distance of an asymptote parallel to the axis of x. 

166. The same function furnishes an illustration of the 
form oo°, namely when x = o. Making the same substitu- 
tion, z = — , we now have 
x 



iog(i+ z n 1 -1 

logy].= J = TT - Z ] =0. 

— ■ 00 I — loo 



Whence 



167. The function 



* 



y = x* 



* This gives the point at which the graph of the function meets the axis of y. 
It is a stop-point since the curve is discontinuous (logjy being impossible) between 
x = o and x = — 1. The curve is again continuous from x = — 1, for which 
y is infinite, to x = — 00 , for which y again approaches the limit e. 



§ XVIII.] ILLUSORY FUNCTIONS OF TWO VARIABLES. 163 
takes the form o° when x = o. In this case, 

log x 



log y\ = x log x] Q = x 



x 

X' 



-1-1 



= o. 



Hence x*\ Q = 1. 

A function which takes this form, or one which takes the 
reciprocal form cc°, is usually found to have the limiting 
value unity.* But this is not necessarily the case, as the 
expression 

a + x 



X l °S x 



will show. This expression takes the form 0° when x = o, 
but it is only another way of writing e a + x ; hence, when x = o, 
its value is e a . 



Indeterminate Forms of Functions of Two Variables. 

168. The value which, as explained in Art. 141, defines 
the ratio of two vanishing quantities is the limit of the ratio 
with which they vanish. Thus, in accordance with Art. 145, 

y — ^ 

the value of the ratio : — , when x and y simultaneously 



x 



a 



take the values a and b, is 

y — b~ 



x — a 



— a, b 



dy 
dx 



_ a, b 



* The reason is that in either case it is the factor log u in the product v log u, 
Art. 164, which is infinite. But when u = o, as well as when u =00, as shown 
in Art. 160, log u bears a zero ratio to an infinite of any finite order. Hence, if v 
is an infinitesimal of any finite order, the product is zero, as in the example above. 



1 64 EVALUA TION OF IND E TERM IN A TE FORMS. [Art. 1 6 8 . 

that is, the value of the vanishing ratio (or ratio with which 
the terms vanish) is, in this case, the ratio of the rates with 
which y and x assume their special values. 

169. When x and y are two independent variables, this is 
a really indeterminate quantity. In like manner, when the 
numerator and denominator are any functions of x and y } the 
value of the fraction, when it takes the indeterminate form, 
depends upon the relative rates of x and y at the instant of 
assuming the special values. For example, the function 

x 2 — 3#v -f- x 
y 2 — xy -f- i 



o 



takes the form— when x= 2 and y = I. Employing the 
usual method, we obtain 



x 2 —'3xy -j- x~ 

f ~ x y + i - 



2x — $y -\- i — 3x 



dy 
dx 



dy 



dxJ Ztl 



2, 



which is indeterminate when x and y are independent. 

170. If, on the other hand, x and y are connected by an 
equation which admits of the simultaneous values x = a and 



y = b f the value of 



dy 
dx 



a, 5 



(or relative rate with which x and y 



assume these values) is determined by the given equation. 
Hence, in such a case, the value of a fraction which takes the 
indeterminate form for these values is determinate. For 
example, in the case of the fraction considered in the pre- 



§ XVIII.] DERIVATIVES OF IMPLICIT FUNCTIONS. 1 65 

ceding article, suppose x and y to be connected by the 
relation 

& + f = 5?> 

which admits of the special values (2, 1) for which the given 
fraction takes the indeterminate form. By differentiation, 
we have 



"I 



dy 2x dy 

-7- = , whence -5— 

dx 5 — 2y dx _ 



_4 



Substituting this value in the expression found above, we 
find the value of the fraction under the given conditions 
to be 6. 

Since, in an example of this sort, y is an implicit function 
of x, the given fraction is virtually a function of the single 
independent variable x. We are thus able to evaluate an 
indeterminate form involving an implicit function. 

Application to the Derivative of an Implicit Function. 

171. When y is an implicit function of x its derivative 
presents itself in the form considered in the preceding article : 
that is to say, as an expression containing y as well as x. 
Suppose now that, for given simultaneous values of x and y, 
the derivative takes the indeterminate form. For example, 
given the equation 

2x 2 y + f + 4*=3> (1) 

from which we derive 

dx y -{- x 2 ' ^ ' 



1 66 EVALUATION OF INDETERMINATE FORMS. [Art. 171. 

Equation (1) is satisfied by the values x = 1, v = — 1, and, 
substituting these values in equation (2), we find 






o 

o 



Hence, applying the differential process, we have 



dy 
dx 



]....=- 



y + x 



dy 
dx 



dy 
dx~ +2X 



dy 



S.1 

<** J,,-« 



1, — 1 






+2 



an equation involving the required derivative in each mem- 
ber. Clearing of fractions, we have the quadratic 



whence 



L^tf-li,-! dx 



= 2, 



»t- x 



Jv 



= — 2 ± 4/6. 



_ii,-i 



Thus the derivative has two distinct values. We infer 
that the curve of which (1) is the equation has two branches 
passing through the point (1, — 1), which is therefore a double 
point of the curve. As x passes through the value 1, the im- 
plicit function y has two values which become equal when 
x = I, but do not become imaginary. Compare Arts. 128 and 
130, which treat of the cases in which one term only of the 
fractional value of the derivative becomes zero. 



§ XVI 1 1.] DERIVATIVES OF IMPLICIT FUNCTIONS. 1 67 

172. When x = o and y = o are simultaneous values of x 
and y, the equation of Art. 168 becomes 



y 

XJ 



_ dy~ 
» dx 



The curve in this case passes through the origin, and if the 
equation is algebraic the value of the derivative at that point 
may be found by evaluating the first member by an algebraic 
process. For example, the curve 

x 2 -\- y 2 — 2x -{- y = o 

passes through the origin. Dividing the equation by x, we 
have 

x-\-y- — 2 -f 2. = o. 

A* AJ* 

Assuming — to have a finite value, this equation becomes, 

X _Jo,o 

when we put x = o and y = o, 

— 2+- =0, .\ - = 2. 

rvJo.o ^Jo,o 

It is obvious that in this process no terms remain, when we 
put x= O and y = o, except those which arise from the terms 
of lowest degree in the original equation. Hence the result 
can be found by simply equating to zero the terms of lowest 
degree. 

173. In like manner, when the equation contains no terms 
lower than the second in degree, the terms of the second de- 



l68 EVALUATION OF INDETERMINATE FORMS. [Art. 1 73. 

gree determine two values of the derivative at the origin, 
which is then a double point. Thus the curve 

y z -f- x 3 — 2>axy = o 

has a double point at the origin, and the tangents at the ori- 
gin are given by 

xy = o; 

that is, the values of tan <p are o and 00 , and the two coordi- 
nate axes are tangents to the curve. If the values of tan 
are found to be imaginary there is no tangent at the origin, 
which is then an isolated point of the curve. 



Examples XVIII. 

1. (cos Ar) cot **, when x = o. 



, F£* 



1 

V*' 



x = o. Ve. 



3. (cos ax) cosec2 P x . x = o. e 2 /3 2 . 

(j \ tan x 
x) > X = °' 



5. (tan x) tan2x , x = \rt. 

6- i-a ) {m > o), x = o. 

JL 

7. (1 -.*)*, x= o. 

8. (sin *) Bec2 *, * = J*r. 



1 
9- •**> at = 00. 1. 



§ XVIII.] EXAMPLES. I09 

10. (sin x) tanx f when x —. o. 1. 

11. (sin x) tanx , x = — . 1. 

2 

.# = o. £ a . 

.*• = o. £"*. 

x = o. 1. 

I 5« (^ 2 ) log ( * + log cos *>, ^ = o. £ 2 * 2 . 

1 



12. 


^;logf 


lime 

a* — x* 


*3- 


(sin 


x yogta.nx^ 


14. 


x* a (a > 0), 






(a + x)* 



l6. X X ~ X y = I. 

17. AT^-1, AT = O. 



I 

e' 
1. 



18. (cos mx)* 2 , x = o. g-i"^ 2 . 

log *\^ 



19. (— ^- ; , x = 00. 1. 



1 



20. (1 ± ^)*, j; = 00 



1. 



l (sinx) imx ( 7e . — V, 

\2 sm 2Jtr/ 



^ = — . 



2 2 m + s ' 



22. If j/ = r , find the value of_y, and also that of—, when 

, — dx 

1 -f e* 

a: approaches zero from the positive and from the negative side. 

dy 
dx 
dy 

dx 

23. If y = r , find the value of y and of Z. when x ap- 

, dx 

1 -f- e* 

dy 

proaches zero from either side. y = °> -y- — ° • 

dy 

' dx 

24. The variables x and y being connected by the equation 

2(1 _ x -\-y) — 4? 2 + 3* 4 = o, 



170 EVALUATION' OF INDETERMINATE FORMS. [Ex. XVIII. 

show that x = o andy = i are simultaneous values, and find the cor- 
responding value of 

y % -f- x z — 8x 2 -\- x — i 



xy 2 — 4X 2 



o. 



25. The relation between x andj' being expressed by the equation 

3^4 _|_ ^3 _ 2 ^ x +J/ ) _j_ 37 _ ^ 

show that x = 1 andjy = 2 are simultaneous values, and find the cor- 
responding value of 

\ A — i6x 

r - * -3' sA- 

dy 
In this example, on substituting the numerical value of -y-, the 

ax 

function again takes the indeterminate form ; it is therefore necessary to 

dy 
substitute the value of '-=— in terms of x andy, and to repeat the process. 

dx 

26. Given x z — axy + (Px — oy 2 + 2 ^ — tf 3 = o; 

show that x = o andjy = tf are simultaneous values, and find the cor- 

r dy 
responding values of — . o and — 1. 

27. Given y* — x* — 4^ 3 -4- 2a 2 *: 2 -j- $ a% y 2 — 2C fy = o; 

show that A' = o and^ = a are simultaneous values, and find the cor- 

dy 
responding values of — * ± 4/2. 

28. Given ^t 4 — lay* — ^a 2 y 2 — 2a 2 x 2 -j- # 4 = o; 

dy 
find the values of — when y = o, also when x = o. 
dx 

«**J±«,o <**Jo,-« <**Jo,*« 

29. Given ^y 3 — a{x -f- a)(x -\-y) == o; 



find the value of 



<**Jo,o" ^Jo,o " 



§ XVIII.] EXAMPLES. 171 

30. Given x* -f- ax 2 y — ay 3 = o; 

dy 
find the values of — when x = o. 
ax 



dy" 

dx_ 

31. If y = •#(.* — 1) log (^ ± \/x), 



o or ± 1. 



dy 
find, by the method of Art. 172, the value of -r- when x =.0; also 

ax 

its values when x = 1, by substituting a?' for .r — 1. 

= — 00 or log 2. 



dy~\ dy 

dxj n ' dx 



10, O ***"' — I I, o 



32. Given y* -J- ^a 2 y 2 — \a 2 xy — a 2 x 2 = o; 

dy 
find the values of -— when x = o and y = o. 

K2 ± 4/7). 

33. Show that the point (a, a) is an isolated point of the curve 

x 3 — Z ax y ~\~J^ -{- a 3 = o. 
-k 34. Show that the point (e, e) is a double point of the curve 

y x = x>. 



CHAPTER VI. 

The Development of Functions in Series. 



XIX. 

Series in Ascending Powers of x. 

I74-. A POLYNOMIAL consisting of a number of terms in- 
volving powers of x with positive integral exponents (including 
the exponent zero) and coefficients independent of x is called 
a rational integral function of x. Thus 

f(x) = A+ Bx+ Cx 2 + • • • + Nx n y 

in which A, B, C, etc., are constants, is the general type of a 
rational integral function. The coefficient A of the zero power 
is the value of the function when x = o and is called the 
absolute term. 

A series of terms beginning with the absolute term A and 
proceeding by ascending powers of x with coefficients follow- 
ing a certain law (so that we can write any number of terms 
we choose) is called an infinite series in ascending powers of x. 
In general, for a given function of x, a series of this kind exists 
which is an algebraic equivalent for the function, and which 
will, at least for certain values of x, furnish a mode of obtain- 
ing the values of the function to any desired degree of accu- 
racy. 



§ XIX.] CONVERGENT AND DIVERGENT SERIES. 173 

The determination of this series for a given function is 
called the development of the function, and the function of which 
a given series is the development is called its generating func- 
tion. 

175. A rational fraction in x (that is one of which the 
numerator and denominator are rational and integral functions 
of x) affords an example in which the development can be 
found by an algebraic process; namely by division, the terms 
of dividend and divisor being arranged in ascending powers 
of x. For example, we thus find 

i — x — X . _ 

- - = I — 2X-\- X 2 — X s -\-X* — • • • , . (i) 



X 



where the law of the successive coefficients is that after the 
second they are alternately -f- I and — i. 

The algebraic equivalence of the series and the generating 
function, which is expressed by the sign of equality in equa- 
tion (i), may be verified by multiplying both members by 
1 -{- x. For this gives in the second member 

I — 2X -\- X 2 — X 3 -\- X 4 — • • • 

-f- X — 2X 2 -|- X 3 — X 4 -f- • • • 



x — X 2 -\-o.x 3 -\-o.x 4 -\- 



which is equal to the given numerator, because we may as- 
sume the value of a series all of whose coefficients vanish to 
be zero for all values of x. 

Convergent and Divergent Series. 

176. It, in equation (i) of the preceding article, we sub- 
stitute for x sl numerical value, we have in the first member a 
value of the generating function and in the second an infinite 



174 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 1 76. 

series of numerical terms. Two very different cases may arise ; 
these we shall illustrate by putting in turn x = i and x = 2. 
In the first place, we have 



= 1 - !+*-* + *- 



We cannot verify this as a numerical equation, but we notice 
that the results of summing 3, 4, 5 and 6 terms of the series 
are J, £, T \, -^, which approach nearer and nearer to ^ the 
value of the generating function. The latter is in fact the 
limit to which these sums approach as the number of terms 
included is increased indefinitely. Under these circumstances 
the series is said to be convergent, and the limit is called the 
sum of the infinite series. 

177. In the second place, putting x = 2 we have the series 

1-4+4-8+ 16- 

in which the result of summing 3, 4, 5, etc., terms are values 
which differ from one another more and more ; the series is 
therefore said to be divergent. The successive sums do not 
approach a limit ; hence there is, in this case, no meaning to 
the expression "sum of the series," and no propriety in 
equating the series to a value of the generating function. 

178. Denote the sum of the terms of the series up to and 
including that containing x n by S n , and denote by R n the dif- 
ference between S n and the value of the generating function. 
Thus we shall have, for any value of x, 

f(x) = S n + R n , 

in which R n is called the remainder ; and depends for its value 
upon n as well as upon x. Then the series is convergent if 



§ XIX.] CONVERGENT AND DIVERGENT SERIES. 17$ 

R n tends to the limit o as n increases indefinitely; and it is 
divergent if R n has no limit. 

In the example considered in the preceding articles, the 
process of division gives us a general expression for the re- 
mainder ; for we have 

1 — x — x 2 , „ x n+1 

= 1 — 2x -f- or — x 6 -f- • • • ± x n qF 



I -f- X I -\-x 

Here, if x is numerically less than unity, R n decreases without 
limit as n increases, that is, it tends to the limit zero when 
n = 00. Hence the series is convergent for all values of x 
between -[- 1 and — 1. These values are therefore called 
the limits of convergence. 

179. In this example, the value of R n increases without 
limit as n increases, for all values of x beyond the limits of 
convergence, and the series is, for such values, divergent. 
At the limits two special cases arise. When x = 1 the values 
of R n are alternately -f- \ and — \. When x = — 1, R n is 
infinite for every value of /z, which indicates that the gener- 
ating function has an infinite value. In each case, the series 
not being convergent is said to be divergent. 

180. Since the successive terms of the series are the dif- 
ferences between consecutive values of R ny they must, in a 
converging series, ultimately decrease without limit as n in- 
creases. But it must not be inferred that the series is neces- 
sarily convergent when the terms so decrease. 

Consider, for example, the algebraic series 

oc + iof+ixs + lx*-] ,. . . . (1) 

of which we at present suppose the generating function un- 
known. If #>i, it is easily seen that the successive terms 



176 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 180. 

will ultimately increase; therefore the series is divergent. 
When x = I, we have the numerical series 

in which the terms decrease without limit. But in this case 
S» can be shown to increase without limit. For, consider 
the terms after the second in groups of two, four, eight, etc., 
ending with the terms J, -J, T \ y etc. ; the sum of the terms in 
each group is greater than -J, and the number of groups is 
unlimited. Therefore the series is divergent. 

181. In fact, unity is for the series (1) a limit of convergence. 
For, supposing x positive and less than unity, every term of 
the series is less than the corresponding term of the series 

X + X 2 -f- X S -f- X 4 -f- • • • . . . . '(2) 

Therefore S H in series (1) is less than S„ in this geometrical 

x 

series. But S n in series (2) has for its limit the value , 

1 — x 

which is the " sum of the series" when x < 1. Therefore S n 
in series (1) must have a limit of less value; that is, the series 
is convergent when x < 1. 

It will be noticed that we can in this way prove the con- 
vergence of any series in which, after some given term, the 
ratio of successive terms is always less than the common ratio 
in some decreasing geometrical series, that is, less than some 
quantity which is itself less than unity, 

182. If the terms of a series, after a certain term, are 
alternately positive and negative and decrease in absolute 
value without limit, the series is convergent. For suppose 
that S n ends with a positive term, then S n + 2 will be less 
than S n because the new negative term is greater than the 



§ XIX.] DIFFERENTIATION OF A SERIES. 1 77 

new positive term. For the same reason S n+i is less than 
Sn+2> hence the values of S n , S n+2 , S n+i , etc., decrease in mag- 
nitude. Moreover, we can show in the same manner that the 
values of S n+1 , S n+S , S n+5 , etc , increase in magnitude. But the 
value of S n+2m , which ends with a positive term, is greater 
than that of S n+2m+ i, which includes the next negative term. 
Therefore each set approaches a limit intermediate to S n and 
S M+1 . But these limits must be the same because the differ- 
ence between S n+2m and S n+2 m+i is a term of the series, and 
therefore by hypothesis can be made as small as we please. 
Thus, for example, the series 

1 — 2T 3 4TI 

obtained by putting x == — I in the negative of series (i) of 
Art. 180 is convergent. 

If, in the same algebraic series, x is negative and numeri- 
cally greater than unity, the terms increase in numerical value 
and the series is divergent. Therefore -|- I and — I are, for 
this series, the limits of convergence; hence in this case we 
have found the series to be convergent at one of the limits 
and divergent at the other. 



Differentiation of a Series. 

183. Let us now suppose that/(x) admits of development 
in the form 

f(x)=A+Bx + Cx*+. . . +Nx« + R H , . . (i) 

in which A, B, C, • • • , N are coefficients independent of x to 
be determined, but R n is an unknown function of x and also 
a function of n. It is assumed, however, that when x = o 



178 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 1 83. 

R n = o, no matter what the value of n. For this reason, 
equation (1) gives, when x = o, 

/(o) = A. 

Thus the absolute term in the development must be the value 
of /(o), just as it is in the rational integral function, compare 
Art. 174. Hence the development is impossible if /(o) is 
infinite.* Of course the assumption that the development 
is possible implies, in like manner, that a finite value can 
be found for each of the other coefficients B, C, etc. 

From the fact that R vanishes with x it follows that every 
series of which the generating function has a finite value when 
x = o must be convergent for some small values of x. 

184. When the development of f(x) is known that of the 
first derivative f\x) can be found. For example, from the 
known development 

= 1 -\-x -\-x 2 -\-x* +rv 4 -{-•••. • • ( 1 ) 



1 — x 

we obtain, by taking derivatives, 
1 



(i-xf 



= 1 -[- 2x -f- $x? -\~4X S -j- • • • . . (2) 



These series are convergent for the same values of x< 
namely for values between -f- 1 and — I. 

185. On the other hand, if the development of f'(x) is 
known, that of f{x) can be assumed in the required form, 
and then the coefficients can be so determined as to make the 

* That is to say, the development is impossible in the form (1). It may happen, 
when/"(o) is infinite, that [x/*(x)] has a finite value, and that xf{x) admits of 
development in the form (1). In that case, dividing by x we should have a de- 
velopment oif(x) in ascending powers beginning with a term in x _1 . 



XIX.] MACLAURIN'S THEOREM. 1 79 



derivative identical with the known series. Thus, supposing 
f(x) = log (i + x), f(x) = (i 4- x)- 1 of which we know the 
development. Therefore, assuming 

log (i + x) = A + Bx + Cx 2 + Dx* + • • • , . . (i) 

we have 

= B + 2Cx + zDx 2 -\ , ... (2) 



I -)- X 

which must be identical with 

= i — x -f- x 2 — x 3 -(-••• . . . . (3) 

1 -\-x 

Hence, equating coefficients, B = 1, 2C = — 1, $D = 1, etc.; 
and, putting # = o in equation (1), A = log 1=0. Therefore 

log (1 -\-x) = x — \x 2 -\- \x* — i% A + • • • . (4) 

Changing the sign of x, we have also 

log (1 - x) = - (x + J* 3 +i* 3 +i*4+ • • • ).* (5) 

Maclaurin s Theorem. 

186, If we assume the development of f{x), as in Art. 183, 
and take successive derivatives, we have the following series 
of developments: 

* Hence — log (1 — x) is the generating function of the series in Art. 180, 
which we found to be convergent for values of x between -f- 1 and — 1. At the 
limit x — 1 the generating function is infinite, while at the limit x = — 1 the series 
is convergent. 



180 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 1 86. 

f{x) = A + Bx + Cx 2 + Dx* + . . • +Nx n + R n , '. (i) 

/"(*) = £ + 2Cx + 3DX 2 + • • • + tiNx"- 1 + — , . (2) 

ax 

f'(x) = 2C-\-3.2Dx+ • • • +w(w-i)AT^- 2 + J 3- (3) 

ax* 



f"(x) = 3.2D + • • • + n(n - 1) (n - 2) Nx»~ 3 + :77 f. (4) 



d# a 



Putting x = o, and making the same assumption as in Art. 
183, we find A =/(0), 5 = /'(o), C - */"((>), £> = ^/"(o), 
etc. Thus the general expression for the coefficient of x n 

in equation (1) is N = —^f n {6). We infer that, if the devel- 
opment be possible, it is 

/(*)=/(o)+/(o)*4-/"(o)^, 

+ f"(o)fr+---+f(°)^+--- ■ (5) 

This result is known as Maclaurin's Theorem. We shall 
give in the next section another demonstration, which de- 
pends upon a single differentiation, and leads also to an ex- 
pression for the remainder. 

187. As an example, we deduce the expansion of e x , 
which is called the exponential series. Putting f(x) = e x , we 
have/ 7 ^) = e x , and in general f n (x) = e x Hence f(o) = 1, 
f(p)= i,/"*(o)=i; and, substituting in Maclaurin's series, 

e* = 1 + x _)_ _ 4- __j_ . . . 



§ XIX.] MA CLA URIN ' S THEOREM. 1 8 1 

This series can be shown, by the method of Art. 1 8 1 , to be 
convergent for all values of x, although when x > I the terms 
begin by increasing. When x = I, we have the following 
numerical series for e: 

I , I 
1 l 1.2 1.2.3 ' 

from which the value of e may be derived with any required 
degree of accuracy. For example, to find e to nine decimal 
places the computation would be as follows : 

2.5 

.16666666667 

4166666667 

833333333 
138888889 

19841270 

2480159 

275573 

27557 
2565 

209 

16 



e = 2.7i828i828 46 

Each term is here derived from the preceding by division, 
and is carried to the eleventh place to insure the accuracy of 
the sum to the nearest unit in the ninth place. 

188. Again, to expand the function sin x in powers of x, 
we put 



l82 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 1 88. 
f{x) = sin x, .'. f\p) = o. 

f(x) = COS #, /'(o) = I. 

/'(#) = - sin #, /'(o) = o. 

/"(*) = - cos x, f"(o) = - I. 

f lv (x) = sin #, / IV (°) — °- 

Since the fourth derivative is the same as the original func- 

tion, we infer that the coefficients of — r in Maclaurrn's theorem 

n ! 

repeat the values o, I, o, — i indefinitely. Hence 

, ••V vV vV . - 

sin* =zx—jY + f\ "~7T + ' " ' ••''•■ W 

In like manner, or from this equation by taking derivatives, 
we find 

zy>2 <y** A**" 

COS X = I - ^j + -y — gy + • - • . . . (2) 

Each of these series, like the exponential series, converges for 
all values of x. 



The Binomial Theorem. 

189. The Binomial Theorem, containing m -f- i terms 
when wis a positive integer, gives the expansion of (a -f- &) w 
arranged in descending powers of a and ascending powers of b. 

When m is fractional or negative, by putting x = - the ex- 

a 

pression becomes a m (i -f- x) m , in which (i -f- x) m is to be devel- 



§ XIX.] THE BINOMIAL THEOREM. 1 83 

oped into an infinite series in ascending powers of x. This 
is readily effected by Maclaurin's Theorem. Thus, putting 

/(*)=(i + *)- .'. /(o) = i. 

f(x)= m(i+x) m -\ /'(o) = m. 

f"{x) = m{m - i)(i + x) m -\ /"(o) = w(w - 1). 

f"{x)= m(m—i){m—2){i-\-x) m -\ /"'(o) = m{m-i)(m—2). 



Hence 

m(m—i) m(m—i)(m—2) 
(i+x) m =i + mx-] — } x 2 + -j x 3 + • • • 

Each term of this series may be derived from the preced- 
ing one by multiplying by a factor of the form 

m — n 

n ~\- 1 

which, when m is a positive integer, ultimately becomes zero, 
thus causing the series to terminate with the term containing 
x m . Otherwise, the fraction ultimately becomes negative, and 
approaches the limit — x as n increases. Therefore, by Art. 
181, the series is convergent when x is between -f- 1 and — 1 ; 
and, by Art. 182, it is also convergent at the limit x = 1. At 
the other limit, — 1, the series is divergent, the corresponding 
value of the generating function being infinite. 

190. In the examples given in the preceding articles, we 
were able to write out any number of terms of the series, be- 
cause the general expression for the nth derivative of the func- 
tion was known. When this is not the case, the labor of find- 
ing the successive values of f'{o), f n {o), etc., may in some 
cases be abridged by expressing the successive functions of x 



1 84 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. I90. 



in terms of the preceding ones; so that, when we put x = o, 
use may be made of numerical values already found. For 
example, if 

f(x) = tan x, f'(x) = sec 2 # = 1 + [f{x\f. 

In finding the algebraic form of subsequent derivatives, we 
shall, for shortness, write f,f r , etc., in place of f(x), /'(#), 
etc. The whole work then stands as follows : 

f{x) = tan x, .\ /(o) = o. 

f{x) = 1 +/ 2 , f{6) = 1. 

fix) = 2ff\ f'(p) = O. 

f"\x)=2ff" + 2f\ /"(o) = 2. 

/"(*) = 2 ff"> + 6/'/", /-(O) = O. 

f(x) = 2ff™ + Zff" + 6/" 2 , /v(o) = 16. 

/ (*) = 2ff*+lOff"+20f l f" i r\o) = O. 

/™(*) = 2jT I +I2/ / / v +30/ ,/ / lV +20/ ,,/2 , 

/ v »(o)= 12x16+20x4=272. 

The process is readily continued, and substituting the results 
in Maclaurin's series we find 

tan* = * + i* 8 + A* 5 + A 7 5* 7 +*«**• + • 

Examples XIX. 

1. Show that, for the series 2) in Art. 184, 

_{n -\-2)x n +* — {n + i)x"+* 
Kn ~ (1 -xf * 

2. Derive an expansion by differentiating equation (2), Art. 184. 

2(1 — x)-*= 1.2 -f 2.$x-\- 3.4^ + 4.5^+ • • • 



§ XIX.] EXAMPLES. 185 

3. Expand log (1 — X -j- x 2 ). 

1 -{-X s 



Put 1 — x -\- x 2 in the form 



1 -f- x 



*y* /y "V*^ -y*» /y»D /y»0 

log (I — X 4- X 2 ) — — X -\ 1 j ... 

2 3453 

4. Develop tan -1 .*: by the method of Art. 185. 

tan l x = x — -J-.* 3 -f- \ x ^ — \ xl + • * * 
7%w is known as Gregory s Series. 

5. Derive the expansion of log (1 — x 2 ) from equation (5), 
Art. 185, and verify by adding the expansions of log (1 -f- x) and 
log (1 — x). 

6. Show that 



and that 



i,i 1 

log 2 = — -| , 

1.2 3.4 5.6 



I — log 2 = 1 (- f- 

2.3 4.5 6.7 



7. Derive the expansion of (1 -f- x)e x from that of e* t 

(1 4- x)e x — 1 -f 2X -f -^- ^ + f- W " r " * .#* -j- 



8. Derive the expansion of (1 -f- x) log (1 -f- x). 

•> -s, -yU -y*4 



1.2 2.3 3.4 

9. Derive the expansion of x tan -1 .*: — -| log (1 -j- x 2 ). 

iy*4t -y* 1 * /yU >V*T 

••V «/V */V 4^V 

1.2 3.4 "'" 5.6 7~8" 



1 86 DEVELOPMENT OF FUNCTIONS IN SERIES. [Ex. XIX. 

10. Find the expansion of e* log (i -\- x) to the term involving 
x 5 , by multiplying together a sufficient number of the terms of 
the series for e x and for log (i -f- x). 

e x log ( i + x) = x -\ 1 f- ^ 1- . . . 

2 3 40 

Derive the following expansions : 



2Jt I 7 x^ 

1 1 . tan 2 * = x 2 A — 1 h • • ■ 

3 45 

e x , x 2 x 3 , 7>x* 1 ijt 5 

12. — - — ■ = 1 H h ^— !-•• 

X 2 KX 4, 6lX 6 

13. sec*=i + — + — +- rf +--- 

* 2 x 4 jr 6 17.* 8 

14. log sec :*: = — - -\ 1 1 ■ 4- . . . 

2 12 45 2520 

„ 2X S , X i 2X 5 

ik. e x sec x = 1 4- •*■ + •** H- 1- h 

3 2 10 

l6. log (i -f- €*) = log 2 H h 



2 8 192 

2 2 Jt: 3 2 4 ;r 5 2 6 x 7 
17. sin .* cos x = x - — 1 ■ — f- 

3! 5 ! 7 ! 



18. COS 2 .* = I — X 2 -\- 



2 3 jr 4 2 5 ^f 6 



4! 6! 

19. (i + 0-=»-[i + a - ' 8 ' 48 

20. Derive the expansion of cos 3 ^, making use of the formula 



, n n(n-\-i)x 2 n 2 (n-\-'i)x z 

1 +-*+ T ■ + 



]• 



cos 3.* = 4 cos 3 .* — 3 cos jr. 



cos 3 * = I — — x 2 4- 3 + 3 JV 4 r- (— i> 3 , ~t, 3 JV 2 « • • • 

2 4.4 ! 1 v / 4(2») ! 

2 1 . Derive the expansion of \/{ 1 -f- sin .*), using the formula 
(sin x -f- cos .*) 2 = 1 -f- sin 2X. 

/y sy/& *%*** *X^ 

a/( i -f- sin x) = 1 -I 5 — i =— r + -7 r + • • • 

T y 2 2 2 .2 ! 2 3 .3 ! 2 4 .4 ! 



§ XX.] TAYLOR'S THEOREM. 1 87 

22. Prove the Binomial Theorem for negative integers by suc- 
cessive differentiation of the series for (1 -{- x)~ l . 



XX. 

Taylor s Theorem. 



|9I. Maclaurin's Theorem gives a development of the dif- 
ference, f{x) — f{o), between the values of a function / cor- 
responding to two values, namely o and x, of the indepen- 
dent variable. Of these values o may be called the initial 
and x the final value of the independent variable. The quan- 
tity whose powers appear in the development is the difference 
between these values, and the coefficients of the powers were 
found to depend upon the initial value. 

We shall now. show that the difference, /(vj — /(%<>), be 
tween any two values of the function may in like manner be 
developed in powers of the difference, x l — x°, between the 
values of the independent variable, with coefficients depend- 
ing upon the initial value x Q and independent of the final 
value x v 

192. If we denote the difference x 1 — x Q by h, so that 

x 1 = x Q -\- h, ( 1 ) 

the development is required of f(x -f- h) in powers of h and 
coefficients depending upon x , that is to say in the form 

f(x + h)= f{x ) + B c h + C Q h? H h NJT + R Q . (2) 

In this equation the coefficients are marked with the suffix 



1 88 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 1 92. 



zero to show that they are functions of x . R oy the remainder 
after the term containing h n , is so marked for the same reason, 
and it is also to be noticed that the assumed form of equation 
(2) makes R vanish with h.* 

Substituting the value of h } we may write equation (2) in 
the form 

f( Xl ) = f{x Q ) + Bo(x x - x ) + C {x x - x o y 

+ D (x x - x Q f H \-N Q {x t - x)» + R . (3) 

If in this equation x be made to vary while x 1 is constant, 
the quantities B Q , • • • , N Q and R Q will be functions of x oi and 
their forms may be ascertained by differentiating the equa- 
tion on this hypothesis. In doing this, it will be convenient 
to replace x by x, thus writing 

/(*,) =/(*) + B{x-x)+c {x- x y • • • + N(x- X y + r, ( 4 ) 

in which B, • • • , N and R are functions of x, while in equa- 
tion (3) B , • - - y N Q and R Q are the special values which they 
take when x = x Q . 

193. Taking derivatives with respect to x, equation (4) 

gives 

o =/'(*) ~ B + (x, - x) ^ - 2C(x 1 -x)+ (x, - xf 2x ' ' ' 

x 1 . / . dN t dR 

-nN(x 1 -x)"- 1 + (x 1 -x)*-^-+-j x . . . (5) 

To render the development possible, B, C, • • • , N and R 

* Compare Art. 183, in which the assumption that R vanishes with x is shown 
to be equivalent to making the absolute term identical with the initial value of 
the function. 



§ XX.] taylor: s theorem. 189 

must have such values as will make equation (5) identical, 
that is to say, true for all values of x. 

It is evident that B may be so taken as to cause the first 
two terms of equation (5) to vanish, and that, this being done, 
C can be so determined as to cause the coefficient of (# — x) 
to vanish, D so as to make the coefficient of (x t — x) 2 vanish, 
and so on. The requisite conditions are 

dB dC 

f\x) -B = o, --2C=--o, ~^ - 3D = o, etc., (6) 

and finally 

N dN dR 
^- Xr dx^+dx=° W 

From conditions (6) we derive 



and in general 



n = ±nx . 

Til 



Putting x Q for x, and substituting in equation (2) the values of 
A o1 B Q , C Q , • • • , N , we obtain 

A*o+h) =Ax )+f(x )h+r(x )^+ • • • +f(xA+R . (8) 

This result is called Taylor's Theorem, from the name of 
its discoverer, Dr. Brook Taylor, who first published it in 171 5. 



190 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 1 93. 

It is evident from equation (8) that the proposed expansion 
is impossible when the given function or any of its derived 
functions is infinite for the value x n . 



Expressions for the Remainder, 

194, To completely satisfy equation (5) as an identity, R 
must be such a function of x as to satisfy equation (7), which, 
after substituting the value found for N, becomes 

dR (x-x) n 



We likewise have the condition, Art. 192, that R = o when 
h = o, that is when x = x v R Q is then the value of R corre- 
sponding to x = x Q . 

It follows that, in order to have a definite value of R Q , it is 
necessary that R should be a continuous function of x for the 
range of values between x 1 and x Q . This requires that its de- 
rivative in equation (9) should be finite and continuous for the 
same range. Hence it is necessary to the existence of equa_ 
tion (8) that f n ^(x) should not become infinite or imaginary 
for any value of x between x Q and x v Since this implies that 
the preceding derivatives oi fix) are likewise continuous for 
the same range, we may state the necessary condition to be 
that f{x) and all its derivatives to the {n -\- i)th inclusive shall 
remain finite and real while x varies from x Q to x -\- h. 

195. Assuming this condition to be fulfilled, various ex- 
pressions for the remainder can be found. These expressions, 
although containing an undetermined quantity, may serve to 
restrict the numerical value of R a between certain limits. 



§ XX.] EXPRESSIONS FOR THE REMAINDER. 191 

For this purpose, we assume a function of x which varies 
continuously from the value unity to the value zero, while 
x varies from x Q to x v For example, 

(x x — x) n+1 

is such a function. Multiplying by R Q , we have the function 

(Xi _ s)»+l 
/ — yPTx ^o , . . . . (IO) 

which has, for each of the extreme values, x and x v the same 
value as R, namely R Q and zero respectively. It follows that 
P — R is a continuous function of x which has the value zero 
for each of the extreme values of x. Hence, as x varies from 
x to x lf P — R starting from the value zero and returning to 
that value must pass through at least one value which is 
numerically a maximum. Therefore the derivative of P — R 
will take the value zero for at least one value of x between x Q 
and x v 

Since x 1 = x Q -f- h, such an intermediate value of x may 
be denoted by x Q -f- Oh, where 6 is a positive proper fraction. 
Hence we may put 

dP-j _ dR 1 

dx Jx +eh dx Jx +0^ ' 

Using the value of P assumed in equation (10), we have 

dP _ (n + 1) (x l — x) n 

dx ~ ~ h n ^ °* 

dR 
Substituting x Q + Oh for x in this and in the value of -=— t 



1 92 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 1 95. 
equation (9), we have, on equating the results, 

This expression for the remainder was first given by La- 
grange. Employing it, equation (8) becomes 

h 2 
A*. + h) =A*o) +f(*o)h +/"(*.) -[+••' 

It will be noticed that Lagrange's expression for the remain- 
der after n + 1 terms differs from the next term of the series, 
simply by the addition of Oh to x Q . 

* Other forms of the remainder result from assuming P in other forms. For 
example, 

/ n (^)-/ n K) ° 

satisfies the necessary conditions, and results in 

*. = <i- 0) n [/ n (^o + *) -/**.)] ^. 

This value of R lies between 

o and [/«K + A)-/«K)I^; 

therefore, see equation (8), /(# o -f ,£) lies between the two expressions 

A*.) +/(* J* +/W n + ' " * + / * ( *° ) ^ 

and 

/M +f(*o) h +f'(x B >^-+ • • • + /•(*. + A ~ • 

See Messenger of Mathematics, New Series, vol. ii, p. 180. 



§ XX.] NUMERICAL VALUE OF THE REMAINDER. 1 93 

(96. The condition given in Art. 194, although necessary, 
is not a sufficient one for the convergence of the series.* But, 
if the series is convergent, the expression for R may be used 
to determine a limit to the error committed in taking S n for 
the sum of the series. 

For example, Maclaurin's Theorem is the result of putting 
x Q = o and h = x in Taylor's Theorem. Hence in the expo- 

nential series, Art. 187, the remainder after the term — r is, 

n\ 

x n+1 

by equation (11), e 9x - r - Thus, in the numerical compu- 

J ^ x J (n-\-i)l 

tation given on page 181, in which x = 1 and n = 14, the 



1 e 



remainder after the term — = is — r > which (because e is less 

14! 15! 

than 3) is less than \ of the last term. It is therefore far too 

small to affect the result. 



Computation by Numerical Series. 

(97. For a given form of the function/", Maclaurin's The- 
orem is the special case of Taylor's Theorem in which the ini- 
tial value is zero. But any development which can be made 
by Taylor's can also, by a change of the form of/", be made 
by Maclaurin's Theorem. For example, if log (1 -f- h) is to be 
developed by Taylor's Theorem, the symbol /"is given the 
meaning log, that is f{x) = log x } and x = 1. But, if we 
change the form of the function, and write F(x) = log(i-|-#), 
we obtain the same development from Maclaurin's Theorem 
(x taking the place of h), while the coefficients, which were 

* The convergence, as we have seen in the preceding section, depends upon the 
character of R considered as a function of n. 



194 DEVELOPMENT OF FUNCTIONS IN SERIES- [Art. 1 97. 

before represented by/(i), f\i), etc., are now represented by 
F(o), -F'(o), etc. Their values would be found thus: 

F{x) = log(i +x) 9 .'. F(o) = o. 

nx) = ^> 2^(0).= 1. 

i?"^) = - (1 + ^)- 2 , ^'(p) = — 1. 

F'"(*) = 2(1 + x)~\ F ff, (o) = 2. 

F l \x) = - 2.3(1 + *)-*, ^(o) = - 2.3. 



Whence, substituting in equation (5), Art. 186, we have 



/y* /v»0 /v* 1 * 



log (l + X) = X -- + ---+... , . . (!) 



which is known as the logarithmic series, and has already been 
otherwise derived in Art. 185. 

198. This series is divergent for values of x greater than 
unity, and is very slowly convergent except for very small 
values of x. For the practical computation of Napierian 
logarithms, a series for the difference of two logarithms has 
been deduced, which may be employed in computing succes- 
sive logarithms; that is, denoting the numbers corresponding 
to two logarithms by n and n -f- h, we require a series for 



n -f- h 

log (n -f- h) — log n = log 

ft 



A series which could be employed for this purpose might be 

ft 1 fa. 
obtained from equation (1) by putting in the form 

ft 



§ XX.] COMPUTATION BY NUMERICAL SERIES. I95 

h 
I -)- — . We obtain, however, a much more rapidly converg- 
ing series by the process given below. 

Substituting — x for x in equation (1), we have 

/V*^ /v**5 /y*4 

. *V vV *V 

log (I — X) = — tf — . . (2) 

Subtracting from equation (1), 

1 -\- x r %* x 5 , x 7 , n 

iog r ^= 2 L x+ r + T + 7 + -"J* • (3) 

a series involving only the positive terms of series (1). 

_ . 1 + x n A- h . . h . 

Putting- = , we derives = ■ — r ; substitut- 

& 1 — x n 2n-\-h 



ing in equation (3), we have 



, n A- h 
log = 2 



h . 1 ¥ 1 W 



+ 7 r~ , M5 + 



_2W + ^ 3 (2W + /0 3 5 (2W -rf- &) 



• (4) 



199. Suppose, for example, it is required to compute 
log 2, it would be quite impracticable to use for the purpose 
the result of putting x = 1 in equation (1), owing to the 
alternate signs, and extremly slow convergence of the series. 
But, if we put n = 1 and h = 1 in equation (4), we have 

log2=2 L3 + 7'3 F+ r? + 7'r + -"J' 

a series which converges with considerable rapidity. 



ig6 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 1 99. 

In making the computation, it is convenient first to obtain 
the values of the powers of \ which occur in the series for 
log 2, by successive division by 9, and afterwards to derive 
the values of the required terms of the series by dividing 
these auxiliary numbers by 1, 3, 5, 7, etc. Thus: 



1 

3 


0.3333333333 = 


1 


0.3333333333 


(if 


370370370 : 


3 


123456790 


ay 


41152263 : 


5 


8230453 


ay 


4572474 


7 


653211 


ar 


508053 


: 9 


56450 


(i) n 


56450 


11 


5132 


(*) 13 


6272 


■ 13 


482 


ar 


697 


: 15 


46 


ay 7 


77 


17 


5 

0.3465735902 
2 



log* 2 = 0.693 14718 04 

200. The Napierian logarithms of the successive natural 
numbers might thus be computed by giving to n the successive 
values 2, 3, 4, etc., and retaining h = 1. But more con- 
venient series are obtained, in some cases, by employing other 
values. 

Thus for log^io, if we put n = 8 and h = 2, we have, since 
log 8 = 3 log 2, 



log, 10 = 3 log, 2 + 



"I + li + IJL+I^L . 

. 3 3 3 5 5 3 9 7 3 13 



] 



The same auxiliary numbers occur as in the computation of 
log 2 above ; thus we have : 



XX.] COMPUTATION BY NUMERICAL SERIES. igj 



X 

3 


0.3333333333 : 1 


o.3333333333 


(« 5 


41152263 : 3 


13717421 


(i) 9 


508053 : 5 


101611 


tt) 13 


6272 : 7 


896 


(i/ 7 


77-9 


9 




3)0.3347153270 






0.1115717757 






0.2231435513 




3 log, 2 


= 2.079441 5412 



log, 10 = 2.30258509 

201. The common or tabular logarithms, of which 10 
is the base, are derived from the corresponding Napierian 
logarithms by means of the relation 



whence 



log,* = log, 10 \og 1Q x, 



log IO * = r— log** = M log,*. 



The constant , , denoted above by M, is called the 

log, re 

modulus of common logarithms. Taking the reciprocal of 

log, 10, computed above, we have 

M = 0.43429448. 



Application to Maxima and Minima. 

202. If the initial value x Q is one of the critical values, 
Art. in, for the function/, we have/ 7 ^) =0, and the series 
for the difference f(x -f- h) —f{pc^ reduces to 



I98 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 202 

h 2 h 3 

Supposing f\x^ to have a finite value, it is obvious that there 
will exist a value of h so small that for all smaller values the 
value of the first term of this expression will be numerically 
greater than the sum of all the others, so that the sign of the 
expression will be that of its first term. In other words, it 
will be that of /"{x^, whether h is positive or negative. Ac- 
cordingly, if f"(x ) has a negative value, the values of f(x -\- h) 
will be less than that of f(x Q ) for values in the immediate neigh- 
borhood of x . Thus/"(# ) will be a maximum in accordance 
with Art. 116. In like manner, a minimum is indicated by a 
positive value oif"(x ). 

203. But, \if"(x ) as well as/'Oo) vanishes, while /"" '(#<>) 
does not, the expression for the difference_/"(x -f- h) — f(x ) will 
begin with the term containing h 3 ; and, in this case, its sign, 
which for small values of h is the sign of its first term, will 
change with the sign of h. Thus the neighboring values to 
f(x ) will exceed f(x Q ) on one side and fall below it on the 
other; so that f(x ) is neither a maximum nor a minimum, 
in accordance with Art. 117. 

Again, if f"{x^) also vanishes, but f 1Y (x Q ) does not, the 
difference begins with the term containing h 4 , and for small 
values of h does not change sign with h) so that the same 
conclusions follow as in the case when f' f {x Q ) is the lowest 
derivative which does not vanish, compare Arts. 118 and 119. 

Evaluation of Vanishing Fractions by Development. 

204. A function which vanishes with x becomes, when 
developed by Maclaurin's Theorem, a series beginning with 
the term containing x or a higher power of x. Denoting this 



§ XX.] EVALUATION OF VANISHING FRACTIONS. 1 99 

power by x n , the function may be expressed as the prod.uct 
of x n by a series of which the absolute term is the original 
coefficient of x n . We thus ascertain at once the power of x 
to which the function bears a finite ratio as it vanishes and 
the value of that ratio. Compare Art. 151, in which it will 
be noticed that the value found for this ratio agrees with the 
coefficient of x n in Maclaurin's Theorem. 

f(x) 
205. When both terms of the fraction — - -; vanish with x, 

(p(x) 

the fraction will have a finite value only when the develop- 
ment of each term begins with the same power of x. Thus, 
the vanishing fraction 

x sin(sin x) — sin 2 ^; -1 
x 6 

will have an infinite value if the numerator is found, on de- 
velopment, to contain a term lower in degree than x 6 , and the 
value will be zero if it contains no term' lower than 'X 7 . It 
is therefore unnecessary, in this case, to retain in the develop- 
ment of the numerator any term whose degree is higher than 
the sixth; and hence, in that of sin (sin x), no terms need be 
retained higher in degree than %~\ Employing the series for 
sin x, Art. 188, we have 

sin (sin x) = sin x — £ sin 3 ^ -j- T -|~o sin 5 # — • • • 

— \ X 6 X TTIo'"' ' ' / 6\ X ~2 X ' ' ') I 120^ " * " 

— *V 3 *v -J- y-jpv j 

whence 

x sin (sin x) = x 2 — J# 4 -f- tV^ 6 * * * » 

also, squaring the series for sin x, 

sin 2 # = x 2 — -Jx 4 + -fax 6 • • • ; 



200 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 205. 

hence 

x sin (sin x) — sin 2 x = ^x 6 . . . 

The value of the given fraction is therefore y 1 ^. 



Examples XX. 

1. Expand e?» + h by Taylor's Theorem, and thence show that 
gx 4- k — e x °e* t . 

2. Expand log (x Q -\-h) t writing the general term and the 
expression for the remainder. 

1 / ^ 1 , h & h* h> 



3. Find the expansion ofy*(^o + ^), when _/"(•%") = x log ^ — x, 

writing the (n -j- i)th term of the series. 

1 /& 2 1 /^ 3 

/j\* -f-^)=:.* log^ — x o + \0gx o .h-\ . . 1 

x Q 1.2 x q 2.3 

I /&* 



x n ~ l [n — 1 )n 

4. Prove that 

5. Prove that 

tan (i?r + ^) = 1 + 2A + 2/* 2 + %h z + JL°£ 4 -j 

6. Compute log 3, and find log 3 by multiplying by the 
value of M (Art. 201). log 3 = 1. 0986123. 

lo g I0 3 = 0.4771213. 



XX.] EX A MPL ES. 20 1 



7. Find log, 269. 

Put n = 2jo = 10 X 3 3 , and h = — 1, 



log 269 = 5.5947114. 



8. Find log* 7, and log, 13. log^ 7 = 1.9459101. 

!°g, *3 = 2.5649494. 
Evaluate the following functions by the method of Art. 205: 
m sin # — sin md 



9- 



^(cos # — cos 7##)' 



1 (1 -\-x)\og (i+x) 

IU. - , 

(x -f- sin- 2 or — 6 sin ^jv) 2 
(4 -f- cos ^ — 5 cos J-*) 3 ' 
tan 7TJt: — 7T^r 
2jf 2 tan 7r.r ' 
<9(2^ + sin26> — 4sin^) 

.3 -j~ COS 20 — 4 COS0 ' 



II. 



12. 



!3- 



when 6 = 0. 


w 




3 


x = 0. 


1 
2' 


.# = o 3 


8.29* 
3 2 * 


.*• = 0. 


7T 2 

~6* 


= o. 


_4 
"3' 



XXI. 

The General Term of the Development. 

206. Maclaurin's Theorem enables us to write the general 

term of the development of a function when the expression 

for the wth derivative is known, as in the simple cases of the 

series for e x , sin x, cos x, log (1 -\- x) etc. Again, putting 

x = o in equation (4), Art. 103, we have for the coefficient 

x n 
oi — r-in the development of e ax cos bx 
n\ 



(a 2 +6 2 )**cos Utan" 1 -). 



202 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 206. 

In the special case where a=b= I, we can assign the 
numerical values, since this expression reduces then to 

7t 

(\f2) n cos n — y 

x n 
which is therefore the coefficient of — : in the development of 

e cos#. The cosine takes periodically the eight successive 
values \/\, o, — |/J, — i, — |/-J, o, |/J, i. The law of the 
coefficients is best seen by separating the terms of even and 
odd degree ; thus 






e* cos x = - 



r_4- + i.6^--64 lTT + 






+ X — 2 — r — A — r + 8 — ; + 



Employment of Differential Equations. 

207. When the general expression for the wth derivative 
cannot be obtained, it may yet be possible to find an ex- 
pression for the particular value which it assumes when x = o. 
This is done by establishing a linear relation between the 
values of successive derivatives, as illustrated below : 

Let it be required to develop sin mz in integral powers of 
sin 2; or, what is the same thing, puttirig sin z = x, to de- 
velop sin [m sin _1 #] in powers of x. Putting 

y = sin [m sin~ 1 x'] f ...... (1) 

we have 

dy _m cos \m sin _1 #] 

dx ~ 4/(1 - * 2 ) ' • ' • ' ( 2 ) 



XXL] EMPLOYMENT OF DIFFERENTIAL EQUATIONS. 20 3 



and 



x 
70 — m 2 sin \ m sin -1 xl 4- m cos \m sin -1 #l sr 

d l y L J ' L J |/( 1 — x 2 ) 



da; 2 



1 — ar 



(3) 



Substituting from equations (1) and (2) in equation (3), we 
have 



d?y dy 



(4) 



a linear relation between ;y and its derivatives. 

Taking the nth derivative of each term of this equation, 
by means of Leibnitz's theorem, Art. 105, we find 



d"+*y 



dx"+ 2 2UX dx^ 1 



— x 



d n+1 y s d n y 

— n(n — 1) 



d^ 1 



y 



dx n 
d n y 



dx"+ l dx" 



, d "y 



-\-m 



dx' 



= 0, 



or 



d n+z y d nJrl y d n v 

(*- x *Ulc^- {2n + l)x a^+ (m2 -^<d = > • • (5) 

a linear relation between any three successive derivatives of 
the given function. 

208. When x — o, this relation takes the simpler form 



n +2 / ,,-| 



(F_ +2 y 
dx 



n+ 2 



1 2 2\ dH y 

= [w — m z ) — - 
J dx n J 



(6) 



^Now, from equations (2) and (3), we obtain 

■jL \ = m y and -? 

dxj dx*_ 



= o. 



204 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 208. 



Hence, putting n equal to I, 3, 5 etc. in equation (6), we have 



dx z _ 



75 -. 
= w(i — m 2 ) y -j~ = m(\ — m 2 )(g — m 3 ) etc., 

CLX __!„ 



and, for all even values of n t 

dx n 



o. 



Substituting these values in Maclaurin's Theorem, we have 
[since /(o) = o] 



sin(msin~ 1 x)=^mx-{- 



or, replacing x by sin z, 



m(i—m 2 ) 3 , m(i — m 2 )(g — m 2 ) 



-x 



sinmz=wsinz 



1 — 



mr — I . 



3! 



sin^- 



5! 



(w 2 — i)(m 2 — 9) . 



**+...; (7) 






5! 



sin 



«*— -.J(8) 



This series will consist of a finite number of terms when m is 
an odd integer. 

In a similar manner, it may be proved that 



cos mz 



m 2 . 9 . m 2 (m 2 — 4) . , 
= I — — j-sin^z -] — sin 4 z , 



2! 



(9) 



the number of terms being finite when m is an even integer. 

209. As another example of a function satisfying a linear 
differential equation,* let 

y = (sin -1 ^) 2 , (1) 

* A direct method of finding the development of the function satisfying a 
given linear differential equation will be found in Differential Equations, p. 166 
et seq.; see also Higher Mathematics, John Wiley & Sons, p. 344. 



XXL] EMPLOYMENT OF DIFFERENTIAL EQUATIONS. 20 5 



then 

dy 2 sin -1 # 



dx |/(i — x 2 )' 
and 



(2) 



1 + 



x sin _1 # 



&y ... 2 V(i - g) /,x 

j* 2 i — ^ * ' " " K5) 

Combining equations (2) and (3), we obtain the differential 
equation 

(i -* 2) S-*£ =2 (4) 

Taking, by means of Leibnitz' theorem, the nth. derivative 
of each term, we have 

, 2 J M+2 y d n+1 y , d n y d n+1 y d"y 

(1 — xr)~ — £= — 2nx _ ., — n{n —i)~- — x- — ^ — n^~ = o, 
J dx n + 2 dx n+1 } dx n dx n+1 dx n 

whence, putting x = o, we derive 



d n + 2 yl 2 d n Y 

- nr 



G- 

-Jo 



(5) 



dx n + 2 J dx n _ 

From equations (2) and (3) we have 

dy~\ _ <Py~\ _ 

dxj Q dx 2 ] Q 

Hence, from equation (5), the odd derivatives vanish, and for 
the even derivatives we have 



do?X ' dx*A a " ' ' dx". 



= 2.2 2 .4 2 , etc. 



206 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 209. 



Finally, substituting in Maclaurin's Theorem, we obtain 
the expansion 

, . 1 X9 V x 2 2 2 X 4 2 2 :d 2 X 6 2 2 .A 2 .6 2 X 8 , "I 

(sin- 1 ^) 2 — 2 — ■ _f ■ -4- -p _U — -i A , 

v J Li. 2 ' 1.2.3.4 ' 1.2. ...6~ 1.2...8 ' 



which may be written in the form 

2 # 4 2.4 # 6 2.4.6 X 



(sin _1 #)^ = 2 



r x 2 



2 + 3 4 + 3-5 6 + 3-5-7 8 



+ 



']. (6) 



210. Differentiation of the result obtained above gives 
the development 

sin -1 ;*; . 2 q , 2.4 _ . 2 4.6 7 . 
_ = x A x 3 A- —^ 5 + — - — x 1 A — • 

1/(1 -**) 3 3-5 3-57 

This is equivalent to a development of the arcual measure of 
an angle in the form of a series involving powers of its sine; 
for, putting sin -1 # = 6, we have 



d = cos 6 sin 6 



1 4. l s in 2 + ^sin 4 A "1, 

3 3-5 J 



Again, if we further transform this by putting x for tan 6, so 

x 2 
that sin 2 # = —7 — ^r, we have 



tan -1 # = 



x 



I -j - X 



I -J-* 2 

- 2 X s 2.4/ X 3 \ 2 , -| 

. I + ^TT^ + 3r5\rT^/ + '"J- (I) 



This series, which was given by Euler in 1793, is convergent 
for all values of x. 



§ XXL] COMPUTATION OF 7t. 20? 



Computation of 7T. 

211. Since tan _1 i = \n y a series for computing n may be 
obtained by putting x = I in the equation just found, thus 

n I 1.2 1.2.3 

2 ^3 3-5 3-5-7 

but this series converges very slowly. 

The earliest computations of n by series were made with 
the aid of Gregory's Series, Ex. XIX. 4, p. 185 : 

tan _1 x = x — %x s -(- \x> — \x^ -| — • 

In this way, Abraham Sharp in 1699 computed n to 71 places, 
using x = -§■ 1/3, whence tan -1 ^ — ^7r. But, in general, much 
smaller values of x were made available by using trigono- 
metric formulae in which \n is separated into two or more 
smaller arcs or multiples of such arcs. Thus, Euler employed 

n 1 . 1 

— = tan -1 — - f- tan -1 — , 
4 2 3 

and Machin computed n to 100 places, using 

7t 1 . I 

— = 4 tan -1 — — tan x 



4 5 239 

Again, Hutton in 1776 employed 

n _ 1 , 1 

— = 2 tan 1 — Htan -1 — . 
4 3 7 

212. When Euler's series, equation (1), Art. 210, is 

used, those values of x which give to the fraction — ; — a deci- 

& 1 4- x 2 



208 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 212. 

mal denominator are particularly convenient. Thus, using 
the formula 

J-= 5 tan-'i + 2tan- 1 ^-,* . . . . (i) 

for x = \, the fraction 

X 2 I 2 



i + x 2 50 IOO ' 
and, for x =-fst 

x 2 9 144 



1 -f-tf 2 6250 100000 

Substituting the equivalent series for the inverse tangents 
in equation (1), and multiplying by 4, we have 



28 

7t = 

IO 



2 2 24/2 



3 100 ' 35 \ioo 



, 3Q336 
~*~ 1 00000 



2 144 24/ 144 y -1 

3 100000 35 Viooooo/ ~*~' "J" 



For convenience of computation, we may write the series in 

* This formula, suggested by Euler in 1779, (like the others which have been 
employed in the computation of %) is readily verified by means of the formula for 
the tangent of the sum of two arcs. Denoting the tangents of the arcs by m and 
n, the tangent of the sum is 

m-\- n 
1 — mn' 

whence, putting m = \, and n = 7 3 ¥ , we obtain 

tan-i i -|- tan-i Y \ = tan-i ^-. 
Hence 

5 tan- 1 ^ -f- 2 tan- 1^ = 3 tan-i \ + 2 tan- 1 T 2 T . 

In like manner the second member may be reduced to tan- 1 \-\-2 tan- 1 ^, and 
finally to tan- 1 1 or \it. 



§ XXI.] COMPUTATION OF n. 209 

the following form, in which each of the letters a, ft, y, etc., 
denotes the value of the term preceding that in which it 
occurs : 



^yr + _4. + i6/_q' 

1 5 L 100 ' io\ioo/ 



6 2a ( 8 2/3 10 2y 

+ 77777; + 



7 1 OO 9 1 00 ' 1 1 1 00 



10112 



1 OOOOO 



288 4 144a 6 144/g 

. ' 1 00000 "■ 5 1 00000 '"7 iooooo 



This form indicates the method by which the value of each 
term is derived from that which precedes it. The numerical 
work for ten places of decimals is given below ; multiplication 
by the factors %, f, etc., is effected by deducting ^-, -J, etc., 
of the quantity to be multiplied. 





3-04 


3- 


a = 


0.00064 


a = O.O0288 


.02a = 


128 


O.OOI44 a = 41472 


\{.02a) — 


182857 


82944 


fi = 


1 097 1 43 


P= 331776 


.02/? = 


21943 


0.00144 /3= 478 




2438 


68 


Y — 


19505 

390 


y = 410 




3.00288332186 




35 
355 


.10112 


d = 


30028833219 




7 


300288332 




1 


30028833 


e = 


6 


6005767 




3.040651 17009 


0.3036515615 




.20271007801 


2.83794109208 




2.83794109208 


7t = 3.I415926536 



2IO DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 2 1 3. 

Non-linear Differential Equations. 

213. The differential equations employed in Arts. 207 
and 209 are the results of eliminating transcendental and 
irrational functions from the immediate results of differ- 
entiation. It is because these equations, see for instance 
equation (4), Art. 209, constitute linear relations between the 
given function and its derivatives, that they give rise to 
simple relations between successive derivatives. 

When the differential equation so found is not linear no 
such simple general relation can be found, but the results of 
successive differentiation may serve to determine the values 
of the derivatives, one after another. For example, given 
the function 

e* * dy e x 

y = e , whence -f- = e . e : 

ax 

combining these equations, we have 

?-*■ • <■> 

which gives the first derivative in an implicit form. Differ- 
entiating again, 



d^ = e \ y+ ic) (2) 

We might obtain a differential equation free from the tran- 
scendental function by combining equations (1) and (2); but, 
in this case, direct differentiation gives a more convenient 
set of equations. Thus we have 

D s y = e x (y + 2Dy + D*y), 

D*y = <?(y + $Dy+ 3D*y + D*y), \ . . . (3) 



§ XXL] NON-LINEAR DIFFERENTIAL EQUATIONS. 2X1 

where the coefficients are those of the Binomial Theorem, see 
equation (2), Art. 107. 

When x = 0, the factor e x ■= 1, and the initial value of the 
function is y = e. Hence by equations (1), (2) and (3), e is 



a factor of each of the quantities y of — 

ax 



2„," 



d 2 y 



, etc., and the 



dx*_ 

coefficients are readily found to be 1, 1, 2, 5, 15, 52, 203, 
etc. Hence the development 

e e* — g[! _|_ x _J_ x * _j_ | X 3 _|_ |^4 _|_ 13^5 _| j # 

214. The process is similar when the original function is 
given in the implicit form. For example, in the equation 

y 3 — 6xy —8 = (1) 

y is a three-valued function of x, of which only one value 
however is real when x = o, having then the value 2. Hence 
as x varies, starting from the value zero, one value of the 
function varies continuously, starting with the initial value 2. 
To develop this value in powers of x, we have, by differentia- 
tion of equation (1), 

(y*-2x)-£-2y=:0; (2) 

whence, putting x = o and y = 2, we find -^ =1, Again 

ax io 

differentiating, 

2y (iJ- 4 '£ +{f - 2x ^ =o; • • - (3) 



d 2 y 
whence, substituting the values already found, — - 

dx 2 ^ 



= 0. In 



212 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 214. 
like manner we can find the values 



d 3 y- 
dx* 



d*y- 
dx" 



1, etc. 



Hence the required expansion is 

y = 2 + X — T \X 3 + ¥¥ * 4 + 

Examples XXI. 



X 



1. Derive the expansion of cot * — from equation (5), Art. 104. 



, X 7t X I JT 3 I X 5 

cot- 1 -^ 1 - + 

a 2 a '3 a 6 S a 



2. liy = sin *x, derive the differential equation 

d 2 v dy 

{1 - X) dx~>- X dx=°> 

and thence the expansion of the function. 



I X Z 1.2 X 5 1.3.5 # 7 

sin- 1 .* = * 4 1 2 ±4 — + 

23 2.4 5 2.4.6 7 



3. Expand e x sin jf. 



„ 2Jf 3 ax* 8x* Sx 7 i6x 9 
e * smx = x+x* + — - — - — -—-+ -—— + 

3! 5! 6! 7! 9! 



4. Expand log [x -j- |/(# 2 -j- jr 2 )] . 



log [*+ V(<* + *")] =loga + --i^ *+ — 4 

&L ' TV ' yj & 'a 2 3fl 3T 2.4 5d D 



§ XXL] EXAMPLES. 213 

5. By means of the series given in Art. 208 derive values of sin $x, 
sin yx, cos 4-x, cos 6x. 

sin $x = 5 sin x — 20 sin 3 :r -f" *6 sin 5 ; 

sin 7Jtr = 7 sin 3: — 56 sin 3 .*: -j- 112 sin 5 *; — 64 sin 7 :*:; 

cos 4.*" = 1 — 8 sin 2 *: -f~ 8 sin 4 .**; 

cos 6jt = 1 — 18 sin 2 .*- -j- 48 sin 4 .* — 32 sin 6 .*. 

6. The function (cos -1 .* 1 ) 2 satisfies the same differential equation 
with (sin -1 -* - ) 2 , see Ex. XI. 17 and Art. 209. Thence expand (cos- 1 .*) 2 . 

Notice that, in accordance with the relation 

n 2 
(cos -1 .*) 2 = {^n — sin -1 .* - ) 2 = 71 s\vr l x -j- (sin -1 .*) 2 , 

the terms of uneven degree give rise to the expansion of sin -1 .*:. 
Compare Ex. 2 above. 

7. Expand \x.-\- \/{d 2 + x 2 )] m . 

?n 2 tniwr — ~l\ 

\x + \/{a 2 + x 2 )~] m =a m -\- ma m ~ l x -| a m ~ 2 x 2 + v '-a m -*x* 

2 - 3- 

m\m 2 — 4) m 4 , , m{m 2 — i)(m 2 — 9) 

4! 5 ! 

8. Given that y = a cos log x -J- b sin log .** satisfies the differ- 
ential equation 

^ dy 

(which is readily verified), derive the expansions by Taylor's Theorem 
of cos log (1 -f- h) and of sin log (1 -{• h). 

h 2 2k 3 ioh* 40/1 5 

C0Sl0g(l+/t)=:I- — + —-— f + — 

sin log (1 + h)=h - _ + _ ^ + §L. 



214 DEVELOPMENT OF FUNCTIONS IN SERIES. [Ex. XXI. 
9. Expand exp (m sin -1 .*). 



m 2 x 2 m(m 2 -f- 1) ^ 3 ( m 2 (m 2 -f- 4) 

^T" 1 3!" 



exp (/rc sin \*) = 1 + w^r-f — - — I ■ 'x 3 -| — p — ~xr -j- 



10. Given the differential equation 

d 2 y dy 

and 

to expand^ in powers of x. 

X 2 x z X* x 5 

^=^ I -^+_ 3 __ 2+( _ s _— +...J. 

11. Expand to five terms the implicit function defined by the 
equation 

e y -j- xy = e. 

X IX 2 I x % 10 X* 

y ~~~' 1 ~~e~ ]r JY? ~*~ J\ (? ~~ 4T & ~* 



XXII. 

Even and Uneven Functions. 

215. Functions of x of which the value is not changed 
when x is changed to — x are called even functions, because 
their developments in integral powers of x can contain only 
terms with even exponents. Functions of which the sign, 
but not the numerical value, is changed when x is changed 
to — x are called uneven functions, because their develop- 



§ XXII.] EVEN AND UNEVEN FUNCTIONS. 215 

ments can contain only terms with uneven exponents. In 
other words, even functions are those which have the property 

/(-*)=/(*); (1) 

and uneven functions are those which have the property 

/(-*) = -/(*) ( 2 ) 

Putting y = fix) y equation (1) shows that the graph of an 
even function is symmetrical to the axis of y; for, if the 
point (a, b) is on the curve, the point ( — a y b) is also on the 
curve. See, for example, the graphs of cos x and sec x, 
Figs. 12 and 14, pp. 63 and 65. Again, equation (2) shows 
that the graph of an uneven function is symmetrical to the 
origin as a centre ; for if (a, b) is a point of the curve, 
( — a, — b) is also a point of the curve. See the graphs of 
sin x and tan x, Figs. 12 and 13. If an odd function is 
continuous through the value x = o, the origin is a point of 
the curve. By differentiating equations (1) and (2), we see 
that the derivative of an even function is an uneven one, and 
that the derivative of an uneven function is an even one. 
This is also evident on differentiating the developments. 

For example, sin x and tan x are uneven functions, and 
their derivatives, cos x and sec 2 #, are even functions. 

216. The development of a function f(x) which belongs 
to neither of these classes may be separated into two series, 
one containing the even and the other the uneven powers 
of x. Denoting the sums, or generating functions, of these 
series by <P(x) and ^(x) respectively, we have 

/(*) = <p( X ) +n») (1) 



2l6 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 2 1 6. 

in which <p and ip denote respectively an even and an uneven 
function. It follows that 

/(-*) = 0(*) - *(*) (2) 

and, combining equations (i) and (2), 

0(*) = *[/(*) +/(- *)] (3) 

«*) = *W*)-A-.*)1- ■,'"••• (4) 

Thus from any function / admitting of development we can 
derive an even and an uneven function defined by equations 
(3) and (4). 

Hyperbolic Functions, 

217. The even and the uneven function thus derived 
from the exponential function &* are called respectively the 
hyperbolic cosine and the hyperbolic sine of x, and are denoted 
by the symbols cosh and sinh.* Thus, putting f(x) = e*, 

cosh x = \(e x -f- e~ x \ (i) 

sinh x = i(e* — e~ x ) ; (2) 

and, from the development of e x y we have 

/y*4 /V" yy*" 

coshx= 1 +ji + Tl + jyH »• • • • (3) 



/v»3 /v**^ «v*' 

— _l__4_ _ 
3'. $ A - 7\ 



sinh * = * + -^ + — f+ryH • . . (4) 



§ XXII.] 



HYPERBOLIC FUNCTIONS. 



217 



The ratio 



sinh x 



is called the hyperbolic tangent of x, and 

is denoted by tanh x; so also the reciprocals of these three 
functions are denoted by sech x, cosech x and coth x. 

218, The graphs of the functions 
cosh x and sinh x are given in Fig. 
31. The dotted lines are the ex- 
ponential curves y = e x and y = e~ x , 
and the ordinate of the curve 
y = cosh #* is the arithmetical mean 
between the corresponding ordinates 
of these two curves. 

From equations (1) and (2), we 
find 




Fig. 31. 



d_ 

dx 



cosh x = sinh x, 



— sinh x 
ax 



cosh x, 



and, since cosh 0=1, the last equation shows that the graph 
of sinh x cuts the axis at the origin at an angle of 45 . 

219. Relations exist between these functions bearing a 
remarkable analogy to those between the circular functions. 
For example, 



* The equation of the catenary curve, the form assumed by a perfectly flexible 
cord of uniform weight attached to two fixed points, is 



y = 



e c — e 



= c cosh — ; 
c 



so that the graph of cosh x is the catenary in which c = I. 



21 8 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 219. 

cosh 2 # — sinh 2 # = 1, 

cosh {x ± y) = cosh x cosh y ± sinh # sinh y, 

sinh (# ± y) = sinh # cosh v ± cosh # sinh y, 

, , , N tanh # ± tanh v 

tanh (# ± y) = ■— — — - — - — r-^- 

1 ± tanh # tanh y 

These formulae are readily verified by means of equations (1) 
and (2) ; they may also be derived from the corresponding 
trigonometric formulae, see Art. 220 below. 

Functions of Imaginary Quantities. 

220. The product of a real quantity x by the imaginary 
factor |/(— 1) is called a pure imaginary quantity. Denoting 
the imaginary factor by i = |/(— 1), we have 

i 2 = — 1 , i B = — i, i 4 — 1 , i 5 = i, etc. 

The function /"(we) of the pure imaginary variable ix may be 
defined as the result of substituting ix for x in the develop- 
ment of f(x). For example, from the developments of cos x 
and sin x, Art. 188, we obtain 

/y>2 /y«4 /y>6 

cos mp = 1 +^j + ^T + 6!+*--> 



sinwc 



= {*+fi+.fr,+-] ; 



hence, comparing with Art. 217, we find 

cos i# = cosh #, sin mp = i sinh #. (1) 



§ X X 1 1 . ] FUNCTIONS OF IMA GIN A RY QUAN TITIES. 2 1 9 

Dividing, we find 

tan ix =5 i tanh x, (2) 

and taking reciprocals, we have 

sec ix = sech x, cosec ix = — i cosech x, 
and 

cot ix = — i coth x. 

By means of these equations, a trigonometric formula 
assumed to hold true for the pure imaginary ix is converted 
into a formula connecting the hyperbolic functions of x. 

221. It is obvious that an even function of the pure imag- 
inary ix must always, as in the preceding article, be a real 
quantity, and that an uneven function of ix must always be 
a pure imaginary. As a further illustration, the development 
of tan -1 #, Ex. XIX, 4, gives 

tan -1 ^ = i(x-\-^x z + %x 5 + ^x 1 -\ ). . . (1) 

Hence, comparing with equation (3), Art. 198, we have 

i 1 -\- x 
tan -1 «e =— log (2) 

2 & 1 — x v J 

Now, if in equation (2), Art. 220, we put v = tanh x, so that 
x = tanh -1 ^, we have 

tan ix =iy\ .*• tan _1 fy = ix = i tanh -1 );. 

Hence equation (2) above gives 

1 I x 
tanh -1 rv = i log (3) 

1 — x VJ/ 



220 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 222* 

222. In general a given function /of the pure imaginary 
ix consists of a real and a pure imaginary part, which can be 
separated by developing the function/. Thus, from the ex- 
ponential series, we find 

e tx = I +IX— - —t—; +-r + Z-T , 

2! 3! ^4!^ 5! 
which may be written 



/y*2 /y»4 jy%f> 

vV *V vV 



2 ! ' 4 ! 6 ! ' 






Here we notice that the real part is the expansion of cos x 
and the coefficient of i is the expansion of sin x; therefore 

e ix = cos x-\-i sin ,x (i) 

Changing the sign of x, 

e -ix __ cos x — ^ s j n x ^ 2 y 

Expressions of this kind, consisting of a real and a pure imag- 
inary part, are called complex imaginary (in distinction from 
pure imaginary) quantities, or simply complex quantities. 
Equations (1) and (2) give 

cos x = i(e ix -\- e~ ix ), 
sin x = — 7 (e ix — e~ ix ), 

in which, by the introduction of complex quantities, the real 
quantities cos x and sin x are expressed in exponential forms. 



XXII.] 



COMPLEX QUANTITIES. 



221 



Complex Quantities. 

223. Denoting any complex quantity by x -f- ^, where 
x and y are real, it is completely represented by the point 
whose rectangular coordinates are x and y, that is to say by 
the position of that point with reference to the origin and 
axes. Let p and be the polar coordinates of the point, so 
that 

x = p cos 6, y = p sin 6 ; 
then the complex quantity may be 
written in the form 

x -\- iy = p (cos 6 + i sin 6). 

Therefore by equation (i) of the pre- 
ceding article we have 

x -\-iy = pe ie . . . (i) 




Fig. 32. 



In this form, p is called the modulus, and 6 the argument of 
the complex quantity. Thus if P, Fig. 32, is the representa- 
tive point (x, y), the modulus is the distance OP of the point 
from the origin, and the argument 6 is the circular measure 
of the angle XOP which determines the direction of OP. 
We have also 



p = ^(oc* -f- y 2 ) , tan 6 = 



x 



When x and y are given, p is always taken as positive, and 
therefore 6 must be so taken that its sine has the algebraic 
sign of v, and its cosine that of x. Thus 6 has but one value 
between o and 27T, and this value may be called the primary 
value of the argument. 

For a real negative quantity, the representing point is on 
the axis of x to the left of the origin ; hence the primary 



222 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 223. 

value of the argument is n. For the pure imaginary i, it is 
\n\ and for — i, it is §7T. 



Conjugate Complex Quantities. 

224. The complex quantity x — iy is called the conjugate 
of x -\- iy. It is represented by the point P f in Fig. 32 situ- 
ated symmetrically to P with respect to the axis of x; and 
we have 

x — iy =p (cos 6 — i sin 6) = pe~ id . . . . (1) 

Thus conjugate complex quantities may be defined as having 
the same modulus and equal arguments with opposite signs. 
The sum of the conjugate quantities x ± iy is the real quan- 
tity 2x, and their product is the positive quantity p 2 . When 
the roots of an ordinary quadratic equation are imaginary, 
they are conjugates. 

The complex quantity cos 6 -(- zsin 0, of which the modu- 
lus is unity, is called a complex unit. A complex unit is there- 
fore the exponential of a pure imaginary, the real coefficient of 
i in the exponent being the argument. Equations (1) and (2) 
of Art. 222 express conjugate units and show that they are 
reciprocals each of the other. 

Functions of the Complex Quantity. 

225. Equation (1) of Art. 223 gives 

log {x + iy) = log p + id, 

in which log p is real because p is a positive quantity. Thus 
the logarithm of a complex quantity is a complex quantity of 
which the real part is the logarithm of the given modulus and 



§ XXII.] DE MO IV RE'S THEOREM. 223 

the coefficient of i is the given argument. We have seen in 
Art. 223 that, for a given complex quantity, the argument 6 
has but one value between o and 27r; but, denoting this 
primary value by 6', 6 admits of the multiple values 2krr -f- 0', 
where k is any positive or negative integer. Thus, 

log (x + iy) = i log (x 2 + f) + i( 2 k7t + &) ; 

whence it appears that the logarithm is a multiple-valued 
function, having the pure imaginary period 2in. The value 
obtained by putting k = o may be called the primary value of 
the function ; for example, the primary value of the logarithm 
of a real negative quantity is the ordinary logarithm of its 
numerical value increased by in. 

226. From equation (1), Art. 223, we have also 

(x -f- iy) m = P m e imd . 

That is to say, to raise a complex quantity to the wth power 
the modulus is raised to the rath power, and the argument is 
multiplied by ra. The modulus is, for this reason, regarded 
as the absolute value of the complex quantity; hence any 
power of a complex unit (as defined in Art. 224) is a unit. 



De Moivres Theorem. 
227. Substituting for the complex units in the identity 

pimQ __ (piQ\m 

their values in the form (1), Art. 222, we have 

cos md-\- i sin md = (cos 6 -\- i sin 6) m . . . (1) 



224 DEVELOPMENT OF FUNCTIONS IN SEEIES. [Art. 227. 



The result is known as De Motive's Theorem. It gives, by 
means of the Binomial Theorem, expressions for cos m& and 
sin md in terms of cos 6 and sin 6. 

The number of terms in each of these expansions will 
be finite when wisa positive integer. Thus, putting m = 3 
in equation (1), we obtain 

cos 3# + i sin 3#=cos 3 # + 3 cos 2 # . i sin 6—3 cos 6. sin 2 — i sin 3 #. 

Whence 

cos 3# = cos 3 # — 3 cos 6 sin 2 #, 

and 

sin 3# = 3 cos 2 # sin # — sin 3 0. 

De Moivre's Theorem also gives us the m\\\ power of the 
complex variable x -\- iy, in the form X ~\- iY, without ex- 
panding the binomial. For 

(x + iy) m = p m (cos 6 + i sin 6) m = p m (cos md -f- i sin md), 

which gives X and Y in finite form, even when m is not an 
integer. For example, 

(2 + fp = 5* (cos I tan -1 J .+ i sin f tan -1 £). 

228. When w is the reciprocal of an integer, say m = — , 
equation (1) becomes 

vYcos # + 2 sin 0) = cos— -4- f sin -. . . (2) 



XXII.] DE MOIVRE'S THEOREM. 22 5 



That is to say, the nth root of a complex unit is found by 
dividing the argument by n. 

Now, if 6 in equation (2) denotes the primary value of 
the argument of the given complex unit, we have seen that 
it also admits of any of the values included in the expression 
6 + 2kn ; hence the argument of the wth root admits of any 
one of the values included in the expression 

6 2klZ 

n n 

Giving to k the successive values 1, 2, 3, ... (n — 1), we 
obtain n — 1 new values of the argument of the root, each of 
which is less than 2n. We have thus n distinct angles, 
namely, 

6 + 2?r (9-J-47T 6 + 2(n — i)tt 

ri n ' n ' n ' 

each of which is the primary value of the argument of a 

1 
distinct value of (cos 6 -f- i sin 6) n . 

229. These angles are derived by successive additions of 

the angle 27t/n. If we continue the process we obtain only 

angles which differ by 27t or a multiple of 2^ from those 

already written, so that they form other values of the 

arguments of which the primary values are those written 

above. Hence we have n, and only n, distinct values of the 

wth root of the given complex unit, namely, 

, . . 6 6> + 27T , . + 27T 

cos \- 1 sin — , cos h 1 sin , • • • 

n n n n 

6-\-2(n—i)7t . . 6-{-2(n— i)n 

cos ■ h 1 sin . 

n n 



226 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 229. 



The geometrical representatives of these n values of the 

nth. root form an equiangular set of 
radii of the unit circle, as in Fig. 33, 
where OP represents the given com- 
plex unit, and 0R V 0R 2 , 0R B , 0R i and 
0R 5 , the five fifth roots of OP. 

230. As a particular case, when 
6 = o, the given complex unit be- 
comes the real unit -|- I, and the ex- 




Fig. 33. 
pressions for the roots becomes 



27t ... lit /\.7t . . 47T 

I, cos ■ \- 1 sin — , cos — -|~ % sin 



n 



n 



n 



11 



2{n— i)n . . 2(n — l)7T 
cos - \- i sin - — — , 



n 



n 



27t 271: 

the last of which may be also written cos — — i sin — . 

n n 

Thus —J— 1 is the primary /zth root of unity ; if the first of 
the imaginary roots is denoted by od } the remaining roots are 
go 2 , oo z , . . . co* -1 , all of which are imag- 
inary, except when n is an even num- 

n 

ber, in which case the root go* has the 
value — I. It will be noticed that 
the roots go and go"- 1 (or go- 1 ) are con- 
jugates, and so also the other imagi- 
nary roots occur in conjugate pairs. 

Fig. 34 gives the geometrical rep- 
resentation of the fifth roots of unity, which are 




-f- 1, cos 72 ± i sin 72 , cos 144 ± i sin 144 . 



§ XXII.] QUADRATIC FACTORS. 22? 

The ?zth root of a complex quantity is an ^-valued func- 
tion of which the several values can be obtained by multiply- 
ing any one of them by the several /zth roots of unity.* 



Quadratic Factors of Certain Algebraic Expressions, 

231. The equation of which the roots are the expressions 
found in Art. 230 is x n — 1 = 0; hence we have 

N / 27T . . 27t\ 

x n — 1 = (x— I) x — cos — — i sin — 
v J \ n n J 

/ \7t . . ^7l\ f 27T . 27T\ 

x— cos 1 sin — • • • [x — cos — 4- 1 sin — . ( 1) 

\ n n j \ n ' n ) 

The product of the factors corresponding to the first and last 
of the imaginary roots is the real quadratic factor 

•> 2 n 

x z — 2x cos (- 1, 

n 

and combining, in like manner, the factors corresponding to 
each pair of conjugate roots, we have, when n is odd, 

/ 2 7t 

X n — I = (X — I) [X 2 — 2X COS + 1 

J \ n 




p 
* In like manner, when m = -, p and q being integers, the function (x-\-iy) m is 

a ^-valued function. But iim is incommensurable, the number of values obtained, 
as in Art. 229, by using different values of the argument is unlimited. In this 
case, we can only deal with the single-valued function corresponding to the pri- 
mary value of the argument of x-\-iy. 



228 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 2yi % 

and when n is even, 

x n — i = (x — i) ( x 2 — 2#cos f- i ) . . . 

\ n ) 

lx 2 —2xcos n 4- \\{x + i). (3) 

232. The roots of the equation 

x 2n — 2X n COS 6 + I = o .... (I) 

are the roots of the two equations 

x n — cos 6 ± i sin 0, 

found by solving it as a quadratic for x n . They are, therefore, 
the n quantities found in Art. 229, together with their con- 
jugates, which are the results of changing the sign of 6. The 
first member of equation (1) is thus separated into 2n linear 
factors. Combining, as in Art. 231, the factors correspond- 
ing to the conjugate pairs of roots, we obtain 

X 2n — 2X n COS 64-1 = (x 2 — 2X COS -+ 1 ) ( XT—2XCOS ^L_j_i ) . . . 

V n )\ n J 

/, 2{n — i)7r + 6 . \ 
lx 2 —2xcos-± '- ! \-l\ . . (2) 

Examples XXII. 

1. Show that log [x -\- \/(i -\- x 2 )~] is an uneven function, and that 
log x -j- /\f{a l -j- x 2 )] is an uneven function increased by a constant. 

2. Prove that each of the following expressions denotes an even 
function of x: 

x cot x and — 1 , 

2 e* — 1 



XXII.] EXAMPLES. 229 

3. Prove that the following denote uneven functions of x: 
1 -f- x 



log 



x / 1 \ 

— , log tan l-7t -f xj 



4. Show that if <p denotes a one-valued function, cp(x 2 ) is an even 
function of x. Compare Ex. I. 22. 

5. Show that an uneven function of an uneven function is an 
uneven function, and that the product of two uneven functions is an 
even one. 

6. Prove the relations: 

sinh 2X = 2 sinh x cosh x, and cosh 2X = cosh 2 x -j- sinh 2 x. 

7. Prove the formulae: 

d tanh x = sech 2 .r dx, d sech x = — tanh .r sech x dx 

d coth ^ = — cosech 2 ^ dx d cosech = — coth x cosech x dx 

8. Find the hyperbolic sine, cosine and tangent of the pure 
imaginary ix. 

sinh (ix) = i sin x, 
cosh (ix) = cos x, 
tanh (ix) = i tan x. 

9. Express the sine, cosine and tangent of the complex quantity 
x -f- iy in the form X -\- iY. 

sin (^ -|- iy) ■= sin .r coshjy -|- i cos ^ sinh y, 

cos (.r -f- ?V) — cos x coshjK — i sin jp sinh jy, 

, , . N tan x sechV , . sec 2 * tanh y 

tan (x+ iy) = -———--—, + , 



1 -[- tan 2 ^; tanh 2 ^ 1 -|- tan 2 x tanhV 

10. Derive the value of tanh _1 jr, equation (3), Art. 221, directly 
from the exponential expression for tanhjy. 

1 I JC 

11. If f(x) = log , prove directly that 

A*)->w=/(r£j> 

also, using the notation of hyperbolic functions, prove the equation by 
means of the relations given in Art. 219. 



230 DEVELOPMENT OF FUNCTIONS IN SERIES. [Ex. XXII. 



12. Prove that 



X x" 

cos m sin -1 f- z'sin ^ sin -1 — 

a a 



Compare the results of putting ix for x in the developments given in 
Ex. XXI. 7 and in Art. 208. 

13. Prove that 

sm~Hx = i log [^+4/(1+ ^ 2 )] = i sinh -1 .*; 
also that 

S inh-'*=log *+* /(a2 + * 8 > . 

a a 

14. Deduce the derivatives of the functions sinh -1 .*; and cosh -1 * 
from formulae given in Arts. 218 and 219. 

—r~ sinh -1 .r = ■ — — -, , -7- cosh -1 .* = 



dx 4/(1 -\- x*)' dx \/{x* — i)' 

15. Prove that the inverse hyperbolic cosine is the two-valued 
function ± cosh -1 .*, where 

cosh -1 .* = log [x -j- \/{x 2 — - 1)]. 

16. Derive the development of sinh -1 .* from that of its derivative 
as in Art. 185. 

smh -1 x = x — £-2. __(_... 

2 3 2 -4 5 2.4.6 7 T 

Compare Exs. XXI. 2 and 4. 

17. Show that sinh -1 — = cosech -1 .* = log - ' ^ — Z # 

18. Develop the function sinh -1 f- log x in ascending powers of 

j, and thence derive a series for sinh -1 .* which is convergent when 
x > 1. 

Sinh -1 .* = log 2.* -| 5- — - -I ^-± ... 

• 2 2X 4 2.44.** 2.4.6 6.* 6 

19. Develop cosh -1 jr in similar form. 

cosh -1 .* = log 2X 5 — — '-^-1 — . — . . 

2 2r 2.4 4x i 2.4.6 6x 6 



§ XXII.] EXAMPLES. 23I 

20. Show, by direct multiplication of the complex quantities a -f- ib 
and c -f- id, that the modulus of the product is the product of the 
moduli, and that the argument of the product is the sum of the argu- 
ments. 

2 1 . Show, by means of the multiple values of log i, that the sym- 
bol \/z represents the real quantity ^/e*, or the product of this by 
an integral power of e 2n . 

22. Find expressions for sin $x and cos $x by means of De Moivre's 
theorem. 

sin $x = 5 cos 4 .* sin x — 10 cos 2 .*: sin 3 .r -f- sin 5 jf. 
cos $x = cos 5 jf — 10 cos 3 .*: sin 2 * 1 -f- 5 cos x sin 4 ;r. 

23. Find expressions for sin mx and cos mx by means of De 

Moivre's theorem. 

m(m — 1) „ , _ 

cos mx = cos m .r 5 ' cos" 1- ^ sm^ 4- • • • ; 

1-2 

_i • m(m — i)(m — 2) _ „ 

sin mx = m cos m l x sm x — — — - cos TO_d jc sin 3 ^ -+-••• 

1.2.3 

24. Find the sixth roots of unity. 

The roots are ± 1, ± J(i ± ^3). 

25. Find the fourth roots of — 1. 

The roots are ± ^2(1 ± i). 

26. Find the cube roots of i. 

The roots are — i, db ^3 -j- 1 i. 

27. Resolve x n -j- 1 into factors. 
When n is even, 



X n -j- I = ( ^ 2 — IX COS f- I j ( 3T — 2JtT COS (- I 

X 1 — 2X COS 71 4- I J ( X 2 — 2X COS 7C 4- I | : 

n j\ n ^ J' 

when n is odd, 

# n -|- I = lx 2 — 2.TC0S — -f- I ) * * * [x 2 — 2X COS 7t -j- I J (x -f- i). 



232 DEVELOPMENT OF FUNCTIONS IN SERIES- [Art. 233 

XXIII. 

The Sine and Cosine as Continued Products. 
233. Putting x = 1 in equation (4), Art. 232, we obtain 

2(1 — cos 6) = 2 W ( I —cos— ) ( I — cos — ) 

47t—6\ ( 2(n—i)7Z-X-6 

I — cos ... I I — cos 



n / \ n 

Using the formula 1 — cos x =2 sin 2 \x, and putting for \Q y 

4 sin 2 = 4* sin 2 *■ sin (- +±) sin 2 Vtl + *V . . 

. ( (fl — I ) 7T , 0" 

sin; ■ -_. 4-— • 

L » w _ 

Supposing < 2;r, and therefore <j> < n, each of the sines 
in this equation is a positive quantity ; hence, taking the square 
root of each member, we have 

sin = 2" x sin —sin —H ---sin L_.-l-.zl . (j) 

n \n nl [_ n n_\ 

If we add 7T to ^>, the first member changes sign ; but, in 
the second member, the first factor assumes the present value 
of the second, the second assumes that of the third, and so on, 

while the final factor becomes sin ( n A — <£), which has the 

\ n / 
present value of the first factor with its sign changed ; there- 
fore the second member also changes sign. Hence equation 
(1) applies, without change of sign, when (j> >-. 



§ XXIII.] THE SINE AS A CONTINUED PRODUCT. 233 

234-. Since the sine of an angle is equal to the sine of its 
supplement, the last factor in equation (1) may be written 

sin ( — — — ). It may therefore be combined with the second 
\n nl 

factor by using the formula 

sin (x -f- y) sin (x — y) = sin 2 # — sin 2 y. 

In a similar manner the third factor and the last but one may 
be combined, and so on; therefore, if n is an odd number, 



sin 6 = 2 n ~ 1 sin -^(sin 2 - — sin 2 -^) 1 sin 2 — 
^ n\ n nj \ n 



sin' 



4> 



(sin s 



n — 1 7r 



— — sin 
n n 



n 



(2) 



Dividing this equation by sin 0, and then making = o, we 
have 



,*-i 



n 



sin 2 — sin 2 
n 



27T 

n 



sin' 



(n — i)?r 



2fl 



■ (3) 



Again, dividing equation (2) by this last equation, we have 



I 







sin d)-=n sin — 
n 



sin 2 —- 
n 



. -It 

sir — 
n 



• 2 0" 

sin^ — 



n 



21t 



sin* 



sin 2 — 



sin 



An—i)7t 



2fl 



* If n were even, the last quadratic factor would take the form 



sin^ 



n — 2 it 



<P 



— sin 2 
2 n n' 

and there would be the single factor cos — outstanding. The corresponding 

n 

factor in equation (3) would be unity, and so also in equation (4), when n is made 

infinite. 



234 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 234. 

Finally, if in the above equation remains fixed while n in- 
creases without limit, we have, on evaluation, 

r 2 "1 r 2 "1 T 0* 1 

sin0 = 0[_i --][_!_— J|_i_— J..., . ( 4 ) 

the number of factors being unlimited. 

235. It will be noticed that this expression for sin con- 
sists of a set of factors one of which vanishes when takes 
any one of the values which make sin = o, namely, 

o, ± 7C, ± 2n, etc. 

Again, arranging the expression in the form 

2 7T — 71 — (p(p7T-\-d)27T-\-(b 

sin = • • . (t\ 

27t 71 1 rt 2n KDJ 

(the series of factors extending to infinity in both directions*), 
we see that changing to 7r -)- 0, and moving each numerator 
one place to the right, the expression is reproduced with its 
sign changed. A second addition of n to the independent vari- 
able restores the original value of the product, thus proving 
the periodic character of the function, that is to say, the 
property sin = sin (2 7r -j- 0). 

* Although the factors both on the right and on the left of in equation (5) 
approach unity as a limit, the product of those on the right (as will be shown in the 
Integral Calculus) is infinite, while that of those on the left has zero for its limit. 
In the deduction of the equation an equal number of factors on each side is taken, 
and that number then becomes infinite. The inclusion of a finite number of factors 
in excess on one side would not affect the value of the product ; but an infinite num- 
ber would. An infinite number of these factors beginning at a point infinitely 
distant has in fact a finite product. Thus it will be shown that if n factors on the 
left and rn factors on the right were taken, and then n made infinite (r having 
a fixed value greater than unity), the product of the extra factors on the right 

would be r™, where m = — . 

it 



§ XXI 1 1.] THE COSINE AS A CONTINUED PRODUCT. 235 

236 A similar expression for cos may be derived from 
equation (4) of Art. 234 by means of the formula 

sin20 

COS = : -,, 

■ 2 sin 
whence 

■*-£)(■-£)(■-£)(■■-£)•••. 



COS 



**l * -%)(*— & 



and, removing common factors, we have 

. r 4 2 ir 40 s ir 40 2 1 



COS 



This equation may also be written in the form 

537 — 2 37T — 20 7t — 20 



cos 



5?r 3?r 7T 

7T+20 37T+20 57r-f-20 



(*) 



, 7T 3?r 5?r 

which exhibits the periodicity of the function, and also the 
values for which it vanishes. 

237. If> m equation (4), Art. 234, we put 0=z%7t y we 
obtain 



l=( i -^n i - 4 

= I I 3 £i 7 

224466 



p-X 1 -^) 



whence 



7T 2 2 4 4 6 6 

2"i 3 3 5 5 7 
which is Wallis's expression for £tt. 



236 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 238. 

238. The coefficient of 3 in the ordinary development 
of sin is — \\ if we equate this to the coefficient of 3 in 
the expansion of the continued product in equation (4), we 
obtain 

n 2 .111 

-5 = l + ~2*~ + y + ^+- • .. . . . (1) 

Again, dividing this equation by 4, we have 

«^ __ I , I , I , ( V 

24. 2* 47 6* 

and, subtracting from the preceding result, 

n * 1 I 1 J 1 I 1 / \ 

8"= I + ^ + ^+7~+*** ' ' • • (3) 

239. By taking logarithms, a continued product is con- 
verted into an infinite series. Thus, equation (4), Art. 234, 
gives 

log sin = log + log ^1 - -j+ log^i - j^j + - • • ; (1) 

whence, expanding the logarithms by means of the develop- 
ment of log (1 — x), Art. 185, and collecting the terms, we 
have 

2 / I I I \ 

log ihT0 = ^ I + r» +?+?+•'•) 

I 4 / Til \ 

+ 2^v I+ ? + ? + 4" 4+ " 7 

I 6 / I I I \ 

+ 3^V + 2« + 3* +4^ / 
+ 



§ XXIII.] BERNOULLI'S NUMBERS. 2$7 

The numerical series in this result are all convergent ; denot- 
ing their values by S 2 , 5 4 , S 6 . . . , the equation may be written 

log^-logsin0=§^+^|^+|^|0«+...*. (2) 

This series is convergent for all values of <p between 
n and — 7T. 



Bernoiillis Numbers. 

240. A series of numbers which occur in the expansion 
of certain functions was introduced by James Bernoulli in 
1687, and has been the subject of much subsequent investi- 
gation. Bernoulli s numbers are the values of the coefficients 

of — r in the development of the function £#cothi#; that 

is to say, putting 

-, e x + 1 
y = i x e* _ 1 ' (0 

they are the values which the successive derivatives of y 
take when x = o. 

These values may be found by means of the differential 
equation satisfied by the function, as in Art. 208. From 
equation (1), we have 

2(ye* — y) — x -f- xe x , 

* It will be shown in Art. 244 that 5 2 = \it 2 (see also Art. 238) and that 
S t = ¥ 1 o7r 4 ; hence this equation becomes 

log <p - log sin = I <p 2 + T £ff 4 -| 

Values of this function, multiplied by the modulus of common logarithms, are 
often given in Trigonometric Tables, to facilitate finding the logarithmic sines of 
small angles. 



238 DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 24O. 



whence, differentiating, 



dy 



dyi 



eX tx + e * y -d L x _ l r I+e *+ xe *>- ■ ■ ^ 



and in general by Leibnitz' Theorem, Art. 105, 



r d m y 
e* 



dx f 



me 



d" l ~ l y 



dx' 



- _| yey — 



d m y -] 
dx m 



tne* -\- xe x . (3) 



Putting x = o, equation (2) gives ^ = 1, and equation (3) 
becomes 



mD m ~ l y + — D m ~ 2 y -\ f- mDy + 1 = -, . (4) 



2 



d r y 
in which Z> r ^ is put for -j— r 



24-f. It is readily shown that y is an even function, hence 
Dy and all the odd-numbered derivatives vanish. It follows 
that two relations can be found connecting any given even 
derivative with lower derivatives; one by means of an even 
value of m and the other by means of an odd value of w. 
Thus, if we put m = 2n -j- 1, we have 

(2n-\- \)2ii(2n — 1) 
(2n+ i)D 2 "y + ^ ; „, } D 2n -*y + . ■ . 



3! 
(2n-\- i)2n _, 2w—i 

+ 2 a* =-—£—; 



(5) 



and, if we put m = 2n + 2, we have 

(2fl+2)(2n+l) (2n+2)(2n+l)2n(2fl-l) r 

2 y* 

(2n-\-2)(2n+ 1) 7 



4 



p2«-2 ;y _|_ 



+ 



D 2 y = n. . . (6) 



§ XXIII. ] BERNOULLI'S NUMBERS. 239 

242. The numerical value of D 2n y is called the nth. Ber- 
noullian number. Since it is found that the values alternate 
in sign while D 2 is positive, the notation adopted is 

B n = X-iy^D*»y (7) 

With this notation, equation (5) becomes 

v 2fl - l n f (2M— l)(2tt— 2)(2tt— 3) D 

B„- n — B n ^+n y^Tl 2 ~ 

■r. , 2W — I 

— (— i) n nB x = — (— i) n — f j — r, (8) 

v J l v J 2{2n + 1) v ; 

and equation (6) becomes 

2»(2»-l) 2»(2»— I)(2»— 2)(2«— 3) 

Bn ~ 3.4 ^ + T^T.^ ^" 2 " * * ' 

2n(2n — i)„ . , w 

1 v J 3.4 2 v / 1 v (W+I)(2W+I) K *' 

Either equation gives, when n = I, 2^ = ^, and when w — 2, 
J5 2 = J-Q. Substituting the value of B x , equation (9)* becomes 

_2n(2n — 1) 2n(2n— i)(2w— 2)(2w — 3) 

B -= 3-4 --- 1 3-4.5-6 5 »-+--- 

2^(2^ — 1) (n — \)(2n— 1) 

-(— l ) n — B 2 +(-ir~ — p-^7 — - v (10) 

3.4 2 1 v / 6(w-f- i)(2w -f~ 1) x 7 

Giving to n the successive values 3, 4, etc., we find 



B 1 -- 


I 

"6' 


£2 


I 

~ 30' 


Bs = 


I 
-42' 


B t = 


I 

"30' 


B 5 ~- 


5 
"66' 






flo = 


691 

> 


Bh - 


7 
— /-) 


B a = 


3617 


-t 










2730 


7 


6 




5IO 







* This equation is the more convenient on account of the recurrence of like 
coefficients. 

■f The values of the Bernoullian numbers up to B S1 have been determined, and 
are given in Crelle s Journal, vol. xx, p. 1 1. 



24O DEVELOPMENT OF FUNCTIONS IN SERIES. [Art. 242. 

The development of the function in equation (1) is, by- 
equation (7), 

X 2 # 4 # 6 ^ 8 



= I +7^-™ + 



12 720 30240 1209600 



The Development of <j> cot <j>. 

243. Putting ix in place of x in the even function just 
developed, we have by Art. 222 

Hx -s-i— = *mp 37: -T- = ia? cot fc. 



2 



e — 1 e* tx — e 



,\ix />—$■*■* 



,y»2 /y.4 -^3 



Hence the development above gives 

ix cot ix = I — #!— — £ 2 ^-- £3^ — 
or putting <£ for £#, 

Another form of the development of cot results from 
taking the derivative of equation (2), Art. 239, which is 

whence 

0COt 0= I-^ 2 -^0 4 -^0 6 • (2) 

244. Comparing the coefficients in these two expressions 
for cot 0, we have, for all values of w, 






(2»)l ~" * 2 * Z * *" 



II ~| 



§ XXIII.] EXAMPLES. 241 



whence 



T.i. A 2 *)"** 



= i + ^+r^ + 



which expresses the sum of any even powers of the reciprocals 
of the natural numbers in terms of Bernoulli's numbers. For 
example, when n = 1, we have the value of S 2 already given 
in Art. 238, and when n = 2, we find 

t 2 4 -f 3 * -t- 4 * -r 90 - 

Equation (1) may also be written in the form 

2(2fl)\ I I ~| 

which shows that the Bernoullian numbers are all positive, 
and also that they increase rapidly with n; for B„ approaches 

2(2n)l 
(but always exceeds) the quantity -, — r^-. 

\ 27t ) 

Examples XXIII. 

1. Show, by means of equation (3), Art. 234, and the result of 
putting 2ti for n in that equation, that 

*_i • o n - 2 3^ • 2 S n ■ z( n — 1 ) 7t ■ 2 ( n — 2 ) n 
1 = 2 sir — sin* 5 ^— sir r_ . . . sir — or sir — , 

271 2fl 271 211 271 

where n is an integer. (Angles in the first quadrant only included : 
thus, for n = 9 and n = 10, 

1 = 2 8 sin 2 io° sin 2 30? sin 2 50 sin 2 70, 

1 = 2 9 sin 2 9 sin 2 27 sin 2 45 sin 2 63 sin 2 8i°.) 

* The logarithms, to ten decimal places, of Bernoulli's numbers up to P 250 have 
been calculated from this formula by Dr. Glaisher, and are published together with 
the first nine figures of their values in the Cambridge Phil. Trans. , vol. xii, p. 386 
and p. 390. 

The full value of B 250 would contain 736 digits before the decimal point. 



242 DEVELOPMENT OF FUNCTIONS IN SERIES. [Ex. XXI I L 

2. Putting (p=\7t inequation (4), Art. 234, prove that the product 
of the even -numbered pairs of factors in Wallis's expression, Art. 237, 

7t 

is the value of , and thence that the product of the odd-num- 

2|/2 

bered pairs is 4/2. 

3. Derive the following continued products : 

71 6 6 12 12 18 18 

3 ~~5" 7" " 13 J 7 19 ' , 

27t 336699 



3V3 2 4 5 7 8 10 

4. Derive equation (3), Art. 238, from the continued product for 
cos 0. 

5. Express the hyperbolic sine in the form of a continued prod- 
uct by putting ix for <p i n equation (4), Art. 234. 



sinh x = x 1 



JC \ / X \ / -v;2 \ 



6. Express the hyperbolic cosine in the form of a continued 
product : 



2 V / , ~-2, 



c dt*=.(i+g)(i + +L)(i+3£ i ) 



JC X 6 X 1 T 

7. Show that = \x; whence, by Art. 242, 

' e — 1 2 e x — 1 

zv* -v* /y-2 *y& -y*0 



e* — i 2 1 '2! '4!' J 6! 






= 1 h _- + 



2 12 720 30240 



§ XXIII.] EXAMPLES. 243 



2X 



8. Separate -5 into partial fractions, and thence by means of 

e — 1 

the result in Ex. 7 find the development of — 



e x -j- 1 



_J_=I-( 22 - I )A I1 + (^-.)^-(^-x)^ r! +- 
9, Obtain a development by adding those of — and 



<?* 4- 1 

g x J I J 

10. Develop, by means of Ex. 8, -— — = 



e x -\- 1 1 -f e~ x e x ~{- 1" 

e x -\- 1 2! 4! 6! 

or,»employing the values of B lt B^ etc., given in Art. 242, 

e? — 1 x x s x 5 i'jx 1 31JV 9 



• • 



e*-J- 1 2 24 240 40320 725760 

£*_[_ j 

11. Derive in like manner the development of from Ex. 7. 

e* — 1 

e x 4- 1 2 „ ^ ^ 3 „ # 5 

— — = h 2B, — — 2B h 2B 

e*- x x ^ x 2! a 4 ! ^ 3 6! 

12. Show that Exs. 9, 10 and 11 give the following developments of 
hyperbolic functions : 

j 2^ — I X 2^ I X* 

cosech x = B y x A £. — i? — 4- • . . 

* l 2 3! 3 3 5! 

1 .r 7^: 3 3ur 5 
x 6 360 15120 

2! 4! 6! s 

1 2 1 2 ^ x 7 7 , 

s 15 315 



244 DEVELOPMENT OF FUNCTIONS IN SERIES. [Ex. XXIII. 



coth X = j -5- X * X 3 4 x b — . . . 

x -2! 4! ' 6! 

I X X 3 2X 5 

" x 3 45 945 

13. Denoting the sum to infinity of the »th powers of the recip- 
rocals of the odd numbers by S' n , show (see Arts. 238 and 244) that 

_ 2 2n_ j (2 2 » - l)7T^B n 

2,1 " 2 2 ' 1 2n ~~ 2( 2 n)\ 

14. Derive an expansion from equation (1), Art. 236, 

2*S' ^ , 1 2^4 „ I 2 6 S* c 
log sec = -^ 0> + -_l 4 + -_i 06 + . . . 

- 2 ( 22 - T ^A 02 , I 2»(2«-l)2? I 2»(2'-l)2?. 

2! ^ T 2 4! ^3 6! ^ ^~ 

15. Express the development of log sin 0, Art. 239, in terms of 
Bernoulli's numbers, and that of log tan by adding log sec 0. 

iB 1 2*B 1 2 5 i? 
log sin = log -±qr -0 4 FT^ — ' ' > 

,o g tan „=,o g 0+^% HJ^Vf^^JVf ■ • 

16. Show, by means of the continued product in Ex. 5, that the 

sinh . ' 

expansion of log — - — consists of the terms of that of log — (Art. 

sin v 

239) with alternate signs changed; and thence, in the notation of 

Bernoulli's numbers, that 

_ sinh0 2*B. , 2 , 1 2 6 ^ 3 ^ 6 , 1 2^B. 10 , 

log — : r- = 2 4 jr- 5 - 6 H r- H • 

6 sin0 2! ^3 6! ^ T 5 10! ^ T 

17. By means of the identity 

tan x = cot x — 2 cot 2X 



§ XXIII.] EXAMPLES. 245 

derive the development of tan x from equation (1), Art. 243, 

2 2 (2 2 - I)B. 2 4 (2 4 - l)tf _ 2 6 (2 6 - i)B % K 

tan x = — - — } —±x-\- — ^ - — J — 2 x 3 + v — ^ — ?Jt 5 -j 



This development may also be obtained from that of Ex. 14. by taking 
derivatives. 

18. By means of the identity 

cosec x = cot \x — cot x 

derive the development of cosec. at, 

1 x 12 3 — 1 _ 12 5 — i ' _. 

cosec x = U ■-— 4- — Bx 6 A — B, x 5 4 «• 

x 6 2 31 3 5! 

19. Derive the expansion of sec 2 * and that of cosec 2 *: by 

taking derivatives. 

, 3 .2 4 (2 4 - i)B. n , 5.2 6 (2 6 - i)B. A 
sec 2 jf = 1 + B. — i '—itf 4- 2 — S_^ ^_3jc4 , 

4! 6! 



3-^ 2 r2 , 5^ 
^ 2 ' ?> 4! 6J 



coseclr = — 4- - 4- £ 2 -x 2 4- £ % x* 4- 



20. By putting = 7rjr and taking derivatives, derive from 
equation (1), Art. 239, the series 



1 1 1 1 1 1 

n cot 7ix = — _j_ -l 4- 1 4- 1- 

X X X — I X -\- I '^—2 X -\- 2 { X — 3 



21. By a similar method derive the series 



ar 1 1 1 1 

— tan nx = 4_ — L 

2 1 — 2x 1 4- 2X 3 — 2 ^ 3 4~ 2X 



246 DEVELOPMENT OF FUNCTIONS IN SERIES. [Ex. XXIII. 

22. By taking successive derivatives of the series obtained in 
Ex. 20 or of that obtained in Ex. 21, and then putting x=.\, derive 
the following numerical series: 

7l 2 III 

^ I I I 

7t* .III, 

96 ' 3 4+ 5 4+ 7 4 ^ ' 

$7T 5 III 



!53 6 3 5 5 5 7 

7T 6 III 



7$e values of the successive derivatives for nx = \n are most read- 
ily found by the process of Art. 190. The first ', third and fifth of these 
results may also be derived from Ex. 13 above. 



CHAPTER VII. 

Application to Plane Curves. 



XXIV. 

Tangent and Normal at a Given Point. 



"& 



245. We have seen that, in the case of a plane curve re- 
ferred to rectangular coordinates, if <fi denotes the inclination 
of the curve at the point (x>y), and s the length of the curve 
as measured from some fixed point on it, we have 

dy • -,. dy dx , x 

tan0 = 5 -, s.n0= s , cos 4> = gp . (i) 

and 

ds = ^idx 2 + dy 2 ) (2) 

See Fig. 18, p. 91, in which the right-angled differential tri- 
angle is drawn. 

246. If x v J\ are * ne coordinates of a known point on the 
curve, the equation of the tangent at that point is found by 
giving to m in the general equation 

y — y 1 = m(x — x x ) 

the value of tan <p at the point (x v v x ) ; hence it is 



dy' 
y - y *=dxA 



{x — x Y ) (1) 



*i».yi 



248 APPLICATION TO PLANE CURVES. [Art. 246. 

In like manner, the equation of the normal at the point 
(x v y x ) is found by giving to the direction ratio m the value 
tan (i^r + 0) = — cot 0; hence it is 



dxi 

y-yi=- 



*if.yi 



dy 

For example, in the case of the ellipse 

x 2 y 2 



(x — x x ) (2) 



dy tfx 

dx a 2 y 

at (x v v x ), a point of the curve, is 



we have — = ^-, therefore the equation of the tangent 



y-y^-^-oc,). 

This equation may, by means of the equation 



a? 



~2 hZ X 



(which expresses that (x v y^) is a point on the ellipse), be put 
in the form 

^1 , m_ _ 

a 2 -r ja — x * 
Again, the equation of the normal is 

y-yi = p^-(* - *i). 

or 

(&y x x — fr^v = (a 2 — h 2 )x x y x . 



XXIV.] TANGENT AND NORMAL AT THE ORIGIN. 249 



Tangent and Normal at the Origin. 

247. When the curve passes through the origin, we 

have seen, in Art. 172, that the value of — - at the origin is 

ax 

the same as that of — , and may be determined by simply 

equating to zero the terms of the lowest degree in the equa- 
tion of the curve. It follows that the equation so found 
is itself the equation of the tangent at the origin, because in 
that line the value of the ratio y.x is constant. Thus, in the 
example given in Art. 172, the tangent at the origin to the 
circle 

x'* -\- y 2 — 2x + y = o 
is the line 

2x — y = o. 

The equation of the normal at the origin (in which m is 
the negative reciprocal of m in the tangent) is, in this case, 

2y -\- x = o. 

Curves Touching one of the Axes at the Origin. 

248. When, not only the absolute term, but one of those 
of the first degree is absent from the equation of the curve, 
it passes through the origin and there touches one of the co- 
ordinate axes. For example, the curve 

x s -\- x 2 y — 2xy 2 + x 2 — 2y = o . . . (1) 

passes through the origin ; and its tangent at that point is the 
line y = o, that is to say, the axis of x. 



250 APPLICATION TO PLANE CURVES. [Art. 248. 

In every such case, a process similar to that employed in 
Art. 172 gives the equation of a simple curve which has a 
much closer contact with the given curve than the tangent 
has. Thus, dividing equation (1) throughout by x 2 , we may 
put it in the form 

y 2 2y 

*+y-2-+i- - 2 = o (2) 



y 

Now we already know that at the origin the ratio — = o; 

x 

V 2 V o 

therefore — =0X0 = 0, but -7; takes the form - and may have 
X X 4 o J 

a finite value at the origin. Hence, putting x = O and y = o, 
we have 



1-% 

x 2 



= 0, 

o, o 



which gives, for the ratio in question, the value £. 
Hence the simpler curve 

X 2 — 2V = O, (3) 

which gives the same value to this ratio, must approach very 
closely to the given curve (1) for small values of x, that is to 
say, in the neighborhood of the origin. 

249, The simple auxiliary curve thus found is in any case 
readily constructed, and is said to give the form of the given 
curve at the origin. In the example above, it is a common 
parabola situated as in Fig. 1, p. 4. Since it lies above the 
axis of x to which it is tangent, we infer that the given curve 
also lies above the axis in the neighborhood of the origin. 



XXIV.] THE PARABOLA OF THE nth DEGREE. 



251 



In like manner, still supposing the given equation to con- 
tain no absolute term, but to contain the term in y, if the term 
of lowest degree not involving y contains x n , we have an aux- 
iliary equation, consisting of two terms only, which deter- 
mines a finite value for the ratio y : x n at the origin. The 
corresponding curve .determines the form of the given curve 
at the origin. 

So also when the given curve touches the axis of x.- 



The Parabola of the nth Degree. 

250. The general equation of the auxiliary curves con- 
sidered above maybe written in the homogeneous form, 



a 



1 y = x r 



M 



The curve represented is called the parabola of the nth degree. 
Supposing a in equation (1) to be positive, the curve passes 
through the point {a, a), as well as through the origin. When 
n>i, the curve touches the axis of x, and when n<i, it 
touches the axis of y. 

The following three diagrams represent forms which the 
curve takes for different values of n 
greater than unity. When n is a 
fraction, it is supposed to be reduced 
to its lowest terms. 

Fig. 35 represents the general shape 

of the curve when n is an even integer, ' — 

or a fraction having an even numerator 

and an odd denominator. 

Fig. 35. 

Fig. 36 represents the form of the 
curve when n is an odd integer or a fraction with an odd 



252 



APPLICATION TO PLANE CURVES. [Art. 250. 



numerator and an odd denominator; the origin is in this 
case a point of inflection. 

Fig. 37 represents the form of the curve 
when n is a fraction having an odd numera- 
tor and an even denominator. In this case, 
y is a two-valued function, and is imaginary 
when x is negative. 

Fig. 35 is constructed for the parabola 
in which n = 4. 

Fig. 36 is the cubical parabola in which 
n— 3. 
Fig. 37 is the semi-cubical parabola in 
which n = f ; the equation being 




Fig. 36. 



or 



a?y 



ay" 



±x s , 



ar 




Fig. 37. 



251. It will be noticed that the curve in 
each case consists of two like branches, sym- 
metrically situated with respect either to an 
axis or to the origin as a centre. Since a was 
assumed positive in equation (1), one of these branches, in 
each diagram, is in the first quadrant. 

The auxiliary equations, found as in Arts. 248 and 249, 
have the more general form 



Ay q + Bx p = o, 



(2) 



where A and B may have either sign, and the positive integers. 
p and q may be both odd, or one odd and one even. (If both 
are even, the equation will indicate an isolated point if A and 
B have the same sign, and will be decomposable, indicating 
more than one branch, if A and B have opposite signs.) The 
three diagrams, in different positions with respect to the axes, 



§ XXIV.] SUBTANGENTS AND SUBNORMALS. 



253 



then give the forms which the given curve * may have at the 
origin. Thus Figs. 26 and 27, pp. 133 and 134, show cases in 
which n < I, or^> < q. The axis which is touched is in every 
case given by the term of lowest degree, and the two quad- 
rants containing the branches are readily determined by the 
odd or even character of p and q. For instance, in 2X -\-y i = O, 
y can evidently have either sign, but x must be negative ; 
therefore the branches lie in the second and third quadrants. 
Again, in 2x -\- y s = o, x and y must have opposite signs; 
hence the branches lie in the second and fourth quadrants. 



Subtangents and Subnormals. 

252. Certain lines connected with a curve and the co- 
ordinate axes, and varying with the point (x, y) of the curve, 
have received special names. The most important of these 
are the subtangent and the subnormal. At the point (x, y),P in 
Fig. 38, let the tangent and normal be drawn, cutting the axis 
of x in T and N t and the ordinate PR = y, then the segment 
TR is the subtangent, and RN is the subnormal. Hence, 
from the triangles TPR and PRN we have, for the subtangent, 

dx 
TR = y cot = y j-, 



and, for the subnormal, 



dy 



RN — y tan = y-f 



dx 



These formulae give positive values 




Fig. 38. 



* When p and q each exceed unity, the equation does not necessarily give the 
form at the origin of a curve in whose equation the terms Ayi -\- BxP are the 
lowest in degree, containing each a single variable. For example, if ^x 2 -j- 2y A 
are the terms of this character, 3X 2 -f- 2y % = o does not give the form at the origin 
if the given equation contains a term in xy. The complete criterion is best applied 
by means of the Analytical Triangle, as explained in works on Curve Tracing. 



254 APPLICATION TO PLANE CURVES. [Art. 252. 

when the direction from T to R and from R to N respectively 
are to the right. 

253. The segments PT and PN are sometimes called the 
tangent and the normal. Their values are 

ds 
PI" = y cosec = y~J~> 

ds 
PN = y sec = y-=-. 

In applying these formulae and others involving ds, it must 
be remembered that equations (1), Art. 245, imply that in- 
dicates the direction in which ds is measured positively. Hence 
the diagram shows that PT when positive is to be measured 
from P in the direction — 0, and PN when positive in the di- 
rection — 90 . 

For example, in the case of the curve 

y = sin x, 
Fig. 12, p. 63, we have 

dy = cos x dx ; 



whence 

and we write 



ds 2 = (1 + cos 2 x)dx z f 



ds = |/( 1 -[- cos 2 x)dx. 



Here, ds being taken with the same sign as dx f <p is the 
direction of the motion of a point moving toward the right. 
Substituting in the expression for the normal, we have 

PN = sin x 4/(1 + cos x). 



§ XXIV.] THE PERPENDICULAR UPON THE TANGENT. 2$$ 

This is positive when sin x, or y, is positive ; accordingly, for 
a point above the axis of x y it is measured in the direction 
— go°, and for a point below in the direction <p-\- go°. 



The Perpendicular from the Origin upon the Tangent. 

254. If OQ in Fig. 39, the perpendicular upon the tangent 
PQ, be denoted by p, we have, from the triangles in the dia- 
gram (which is so drawn that x, y, sin and cos are posi- 
tive), 

^ Y 

p = x sin — y cos <p. 

Substituting the values of sin <p and 
cos 0, this becomes 

xdy — ydx xdy — ydx 



ds 



V{dx* +dy 2 )' 




/ 

The figure shows that the direction IG ' ^ 9 * 

of OQ (or p as drawn from the origin) is — 90 when posi- 
tive. 



Curve Tracing. 

255. A curve whose equation is given is said to be traced 
when the general form of its various branches, and their posi- 
tion with respect to the coordinate axes, is determined. We 
notice, in the first place, certain forms of symmetry of which 
the occurrence will be indicated by the form of the equation. 
First, when it contains only even powers of one of the coor- 



256 APPLICATION TO PLANE CURVES. [Art. 255. 

dinates the curve is symmetrical to one of the axes. Thus 
the curve 

*(* 2 + f) + <*(* 2 - f) = o . . . . (1) 

is symmetrical to the axis of x ; because, if the equation is sat- 
isfied by the point (x, y), it is also satisfied by the point (x> — y) 
situated symmetrically to (x, y) with respect to this axis. 

Again, if every term is of an even degree with respect to 
x and y jointly, or if every term is of an odd degree, the curve 
is symmetrical with respect to the origin as a centre. For 
example, 

Ax 2 + Bxy + Cf = D 

is thus symmetrical, because, if (x, y) satisfies the equation, 
(— x, — y) also satisfies it. This is in fact the equation of the 
conic with respect to the centre as origin ; if B = o, it is the 
conic referred to its axes, which is symmetrical to both axes 
and therefore also to the origin. 

256. If the equation can be solved with respect to one of 
the variables, so as to make it an explicit function of the other, 
it is generally advantageous to do so. Thus, if the equation 
is put in the form y=f(x), the curve becomes the graph of 
a known function, so that, by assigning values to x } we may 
determine as many points as we choose through which to 
draw a continuous curve. 

The most important things to be determined are the limits 
of continuity, whether indicated by infinite values of y as illus- 
trated in Fig. 3, p. 6, or by values of x on one side of which 
y is imaginary, as in Fig. 4, p. 7. Next to these come the 
values of x for which y = o, giving points where the curve 
cuts the axis of x, and the values of y corresponding to 
x = o and to x = 00 respectively. 



XXIV.] 



CURVE TRACING. 



257 



257. As an illustration, let us take the curve represented 
by equation (1) of the preceding article, which is known as the 
Strophoid. This curve is symmetrical to the axis of x ; solving 
its equation for v 2 , we have 

Y 



y% = 



x^ix-^a) 
a—x 



(1) 



Here y =0 when x = — a and when 
x = o : v is infinite when x = a, and 
is real only between the limits x= ±a. 
Hence the general shape of the curve 
is that given in Fig. 40, consisting of a 
loop between x = — a and x = o and 
infinite branches between x = o and 
x = a. The tracing indicates the 
existence of a maximum ordinate 




Fig. 40. 

This corresponds to the 
maximum value of y 2 in equation (1). Taking derivatives 



dy __ (a — x)(2ax -\- 3^)+ ax 2 -\-x 3 _ 2x(a 2 -{-ax—x 2 ) > 



dx 



{a — xf 



hence y 2 is a maximum when x 2 



ax = a 2 . 



(a — x) 2 
The roots of 



a, 



this quadratic are x = — (1 ± j/5). The positive root is be- 
yond the limits of real values of y\ the negative root is about 
— .6a and the corresponding value of y is almost exactly .3a. 
These are the coordinates of A in the diagram. The tangents 
at the origin are found, by the method of Art. 247, to be the 
lines y = ± x, bisecting the angles between the axes. 

258. The maxima and minima values of either coordinate 
are its limiting values when it is made the independent vari- 
able. Consider, for example, the curve 

v 4 + x 2 — f — o, 



2 5% APPLICATION TO PLANE CURVES. [Art. 258. 

which is symmetrical to both axes and therefore also to the 
origin as a centre. Solving for x, 

x= ± y \/(i — y 2 )\ 

hence the limiting values of y are ± 1 (the corresponding 
points are those in which the curve cuts the axis of y), and x 
is real between the limits. The maximum value of x may 
be found either by the differential method, or as follows: 
Solving for y, we have 

whence the limiting values of x are ±£, and y is real between 
the limits. These are therefore the numerically greatest values 
of x, and the curve is limited to the rectangle between the 
lines}' = ± I and x = dz i. The limiting values of x make 
y 2 = £, hence the curve touches the sides of the rectangle in 
the points (o, ± 1) and (± -J-, ± J 4/2). It passes through the 
origin at angles of 45 ° with the axes, and resembles in form 
a figure 8. 

Points of Inflexion, 

259« The curve considered above obviously has points of 
inflexion at the origin. In other cases, the form of the curve 
may indicate the existence of points of inflexion which, when 
the equation is solved for one of the variables, may be found 
by equating the second derivative to zero, see Art. 99. As 
an illustration, let us take the curve 

y = , 

y a — x 



§ XXIV.] EXAMPLES. 259 

which is symmetrical to the axis of x, and in which y is real 
only between the limits x = o and x = a. It cuts the axis of 
x only at the origin, where it touches the axis of y (as indi- 
cated by the double root when we put x = o). Therefore 
tan must be infinite at the origin and must at first decrease; 
but it is again infinite when x = a, because this value of x 
makes y infinite. There is therefore at least one point on the 
positive branch of the curve where the slope is a minimum, 
that is to say, a point of inflexion. The location of this point 
(of which the abscissa will be found to be Ja), together with 
the slope of the curve at that point, will determine its form 
with considerable accuracy. The curve is known as the Witch 
of Agnesi. 

The methods explained in the next section, which are ap- 
plicable when the equation cannot be solved for either vari- 
able, are also often useful even when it can be so solved. 



Examples XXIV. 

1. In the case of the parabola of the nth. degree 

find the equations of the tangent and the normal at the point (a, a). 

2. Find the equation of the tangent at any pointof the curve 

x n + y H = & n - 

yy*~ x -J- xx*- 1 = a H . 

3. Find the equation of the tangent at any point of the curve 

x m y H = a m+H . 

nx x y + m y x x = ( m + n ) x x yv 



260 APPLICATION TO PLANE CURVES. [Ex. XXIV. 

4. Show that all the curves represented by the equation 

(different values being assigned to n) have a common tangent at the 
point (a, b) ; find the equation of this tangent. 

5. Show that the equation of the tangent to the curve 

x* jr _ 
cr b 

at the point (x x , y x ), is 

crx^y -j- $y*x = arb*x x y x * ; 
and, denoting the intercepts on the axes by -%" and^ , prove that 

x -l +-*L.= 1 



6. Given the curve 



x 2 — 2y 2 — \xy — x = o, 



show that the point (1, — 2) is on the curve, and find the equation of 
the tangent line at this point. 

7. Find the subtangent and the subnormal of the parabola 



y 2 =4ax; 



also the value of/ in terms of x. 

x\/a 



For the upper branch, p = — 



8. Find the subnormal of the ellipse 

a 2 ^ b 2 ~ 



§ XXIV.] EXAMPLES. 26l 

9. Prove that the normal to the catenary curve 

c ( * -* 
y— -ie c -\-e c 

y 2 
(see Fig. 31, p. 217) is equal to — . 

c 

10. Trace the curve y = x(x 2 — 1), finding the maximum ordinate 
and the slope at the intersections with the axis of x. 

11. Trace the curve jy = x(i — x) 2 , finding the maximum ordinate 
and point of inflexion. 

12. Trace the curve ay 2 = x{a — x) 2 . 

13. Trace the curve x 2 y -(- a 2 y — a 3 = o, finding points of inflexion. 

14. Trace the curve xy 2 = x 2 ~{-y 2 . 

15. Trace the curve y 2 = x z — x*. 

16. Trace the curve y(x — a) 2 = a 2 x, finding a minimum ordinate 
and a point of inflexion. 

17. Trace the' curve (y — x 2 ) 2 = x 5 . 

18. Trace the curve x{x 2 -\-y 2 ) = 2ay 2 , which is the Cissoid of 
Diodes. 

19. Show that in the curve y 2 =/(.*:) the abscissa of a point of in- 
flexion will satisfy the equation 

20. The equation of the Conchoid of Nicomedes is 

(x 2 -\-y 2 ){x — a^ 2 = b 2 x 2 : 

trace the curve in the three cases when b < a, b = a, and 3 > a. 

The maximum ordinate is (b l — d*f ; the abscissae of the points of 
inflexion satisfy x z — 3a 2 .*: -f- 2a(a 2 — b 2 ) = o. 



262 APPLICATION TO PLANE CURVES. [Art. 260. 



XXV. 

Points at Infinity, 

260. When the equation of a curve permits one or both 
of the coordinates to take an infinite value, that is to say, to 
increase without limit, the curve is said to have a point at 
infinity. If, while one of the coordinates becomes infinite, 
the other remains finite and has a definite limiting value, there 
is a straight line, parallel to one of the axes, to which the 
point describing the curve approaches without limit as it 
recedes to infinity. This line is called an asymptote. We 
have had examples in Figs. 3, p. 6; 4, p. 7; 10, p. 57; 
17, p. 71, etc., where a finite value of the variable regarded 
as independent gives an infinite value to the function ; or else, 
as the independent variable becomes infinite, the function 
approaches without limit to a definite value. 

dy 

In these cases, -7- tends to one of the limits zero or infin- 
ity; and, if the point of contact of a tangent line recedes to 
infinity, the tangent approaches the asymptote as its limiting 
position; hence the asymptote is called the tangent at infinity. 

On the other hand, Fig. 12 exemplifies a case where, as x 

dy 
becomes infinite, neither y nor ~ approaches a definite limit; 

there is then no asymptote and no definite tangent at infinity. 

261. In the cases considered above, the point at infinity is 
said to be in the direction of one of the axes. The direction of 
the point at infinity is, of course, the direction of the line join- 
ing it to the origin. Hence, just as m is called the direction 



§ XXV.] POINTS AT INFINITY. 263 

y 

ratio of the line y = ntx, so the value of the ratio — at the 

x 

infinite point gives the direction of the point at infinity. 
When x and y become infinite simultaneously, the ratio 

- takes the form — and may have a finite value. If so, 

x 00 

giving this value to m, the point at infinity is in the direction 
of the line y = ntx. The ratio y : x at infinity is the same 
for all parallel lines of the form y = mx -\- b, therefore parallel 
lines are said to pass through the same point at infinity. If 
an infinite branch of a curve has an asymptote in the direc- 
tion y = mx its equation will be of the form y = mx -\- b. It 
must not be inferred that an asymptote necessarily exists, 
since for this purpose it is necessary that b should admit of a 
finite value. See Art. 271. 

262. To illustrate the method of finding the points at in- 
finity for an algebraic curve, let us take the curve whose 
equation is 

x 3 — xy 2 + ay 1 — a 2 y = o (1) 

Dividing through by x 3 , we have 



jy 2 ay 2 a 2 y 



Putting x = 00 and y = 09, while assuming that the ratio 

— has a finite value, we find 
ad*. 



264 APPLICATION TO PLANE CURVES. [Art. 262. 



whence — 
x. 



= ± I, which determines two points at infinity in 



the directions of the lines y = ± x. 

263. It will be noticed that, in this process, all the terms 
except those of the highest degree in the given equation dis- 
appear from the result, which is therefore the same as if the 
equation had consisted only of the group of terms of highest 
degree, namely, x 3 — xy 2 = o. In fact, writing this equation 
in the factored form 

x(% +y)(x — y) = o, (3) 

we see that the factors which, separately equated to zero, give 
its several roots constitute the equations of the lines through 
the origin in the direction of the several points at infinity. 
The root x = o, in equation (3), corresponds to a point at 
infinity in the direction of the axis of y; for this point, 

x v 

the ratio — is zero, or — is infinite. 
y x 

The number of points at infinity may, as in this case, equal 

the index of the degree of the equation, but cannot exceed it. 

If some of the roots of the equation are imaginary, there are 

fewer points at infinity, and when the degree is even there may 

be no real points at infinity. 



The Equation of the Asymptote, 

264. We proceed to determine the position of the asym- 
totes corresponding to the points at infinity determined by 
equation (3). For this purpose, we write the group of terms 
of the highest degree in the factored form. Then, for the 



§ XXV.] THE EQUATION OF THE ASYMPTOTE. 26$ 

asymptote corresponding to x — y, divide the equation of the 
curve through by the other factors, thus : 

x y- x{x + y) (4) 

Now, when a point recedes to infinity on this branch of the 
curve, x and y increase without limit, but with a limiting 
ratio of equality; thus the first member takes the illusory 
form oo — oo. But the second member, which takes the form 
oo /oo, may be evaluated as in Art. 155. Thus, dividing 
both terms of the fraction by x 2 y we have 



x — y 



in which, puttings =00, we have, since— = 1, 

X^iao 



x — y = — \a (5) 

We infer then that the distant points on the curve, both in the 
first and in the third quadrant, approach without limit to the 
corresponding points of the line represented by this equation. 
This line is therefore an asymptote.* 

* The value of the expression x —y -f- \a, which is zero for a point on the line 
(5), is for a point on the curve its vertical distance from the asymptote (below it 
when positive, and above it when negative), and this distance approaches zero as a 
limit. 



a 2 


y 


/ 
-^ 


X 


X 




1 + 


y. 

X 



266 



APPLICATION TO PLANE CURVES. [Art. 265. 



265. It will be noticed that, in this process, only the terms 
of highest degree in the numerator (which are those of the 
next to the highest degree in the equation of the curve) can 
affect the result. 

In practice, it is unnecessary to reduce the fraction in the 
second member to the complex form. Thus, in finding the 
asymptote corresponding to the factor x -f- y, we write 



x-\- y 



ay 1 



x{x - y) J x m _ y m „ 
Again, corresponding to the factor x, we have 



a 

2 



• (6) 



X = 



ay 



2 n 



x 2 — y 2 j 



a; 



(7) 



. . . x- 

since in this case — 

y. 



= o 



266. In this last case, it will be noticed that the asymp- 
tote depends solely upon the terms containing y 2 in equation 
(1), namely — xy 2 -f- ay 2 . Thus the absence of the term con- 
taining y 3 indicates a point at infinity in the direction of the axis 
of y, and then the equation of the asymptote is found by equat- 
ing to zero the coefficient of y 2 , that is to say, it is — x -J- a = o. 



Tracing of Curves with Infinite Branches, 

267. The construction of the asymptotes, when they exist, 
is of paramount importance in tracing the general form of a 
curve. For example, in the case of the curve 



x 3 



xy 4 -J- ay — ay = o 



(1) 



XXV.] CURVES WITH INFINITE BRANCHES. 



267 




considered in the preceding articles, the three asymptotes, 
equations (5), (6) and (7), Arts. 
264 and 265, are constructed as 
dotted lines in Fig. 41. These 
lines, together with a few actual 
points of the curve, such as its 
intersections with the coordinate 
axes, will generally serve to de- 
termine the shape of the several 
branches of the curve. 

268. In the present case, 
putting z = O in equation (1), 
we have ay 1 — a 2 y = o, whence 
y = o and y = a, showing that 
the curve passes through the 
origin and through the point 
(o, a), the point A in Fig. 41. 

Putting y = O in equation (1), 
we have x 3 = o, showing that the curve meets the axis of 
x only at the origin. The tangent at the origin is this axis, 
and, by the method of Art. 248, the form at the origin is 
given by 




Fig. 41. 



XT 



ary = o. 



Here x and y have the same sign (see Art. 251); hence the 
branch passing through the origin has the form indicated in 
the diagram. 

269. Since the curve is of the third degree, a straight 
line will in general cut it in three points. But special cases 
arise : for example, in the present case all three intersections 
with the axis of x coincide at the origin, because that axis 
is a tangent at a point of inflection. Again, the axis of y 



268 APPLICATION TO PLANE CURVES. [Art. 269. 

cuts the curve in but two finite points, the third intersection 
being at infinity, because that axis is parallel to an asymp- 
tote. 

Again, the asymptote itself, being a tangent at infinity, 
has but one finite or actual intersection with the curve. Ac- 
cordingly, puttings = a in equation (1), we have a?—a 2 y = o, 
giving the single root y = a which determines the point (a, a) r 
the point B in Fig. 41. 

In like manner, the asymptote x — y = — \a will be found 
to cut the curve in the single point { — a, — \a), C in Fig. 41. 
It is clear that the other asymptote cuts the same branch of 
the curve, therefore the branch through A cannot cut either 
asymptote, and must approach the upper ends of these asymp- 
totes in the manner indicated. Furthermore, the lower 
ends of these asymptotes must be approached by a third 
branch, as represented in the diagram. 

Maximum and Minimum Coordinates. 

270. The tracing of this curve shows that a point of 
minimum ordinate must exist in the branch through A, and 
one of maximum ordinate in the lower branch. To deter- 
mine these we have, by differentiation of equation (1), Art. 
267, 

(33c 2 — y 2 )dx — (2xy — 2ay -\- a 2 )dy = o. 

dy 
It follows that — = o when 
dx 

3X 2 — y 2 = o, or y = ± x 4/3. 

Therefore the horizontal points of the curve are its intersec- 
tions with these straight lines which pass through the origin. 



§ XXV.] PARABOLIC BRANCHES. 269 

Fig. 41 shows that the line y = x ^3 (which makes an angle of 
6o° with the axis of x) has no other real intersection with the 

curve. But, putting x = — — in equation (1), we have the 

cubic 

2v 3 

whence the three values of y are zero (corresponding to the 
origin) and the roots of the quadratic 

These are y = .77a and y = — 3«37#> tne former being the 
minimum ordinate at D and the latter the minimum negative 
ordinate at E in the lower branch. 

Parabolic Branches. 

271. When the factor of the group of terms of highest 
degree, indicating a point at infinity, is of the second degree, 
the process given in Art. 265 for the equation of the asymp- 
tote results in an infinite value for the second member,* show- 
ing that the point on the curve recedes indefinitely from the 
straight line drawn through the origin in the direction of the 
point at infinity. A branch of this kind, of which the common 
parabola presents the earliest instance, is said to be parabolic. 

212 u In the case of the curve 

2x 2 y+ y 2 + 4x= 3, (1) 

the single term of highest degree, 2x 2 y, indicates a point at 
infinity in the direction of each axis ; and the case considered 

* Except when the group of terms of next highest degree contains the same 
factor, or is absent from the equation, in which case there are two parallel asymp- 
totes. See Exs. 4, 7, etc., below. 



270 APPLICATION TO PLANE CURVES. [Art. 272. 

o , - 

above arises with respect to that in the direction of the axis 
of y. That is to say, there is a branch upon which, as the 

y 
point recedes indefinitely, — becomes infinite, but x becomes 

x 

infinite as well as y. In this case, x 2 will be found to have a 
finite ratio to y. For, equation (i)can be put in the form 

"XT , 1 X *\ , . 

h-i +4- 2 = - a , (2) 

y y y 



which, when y is infinite, reduces to 

2X 2 ' 

y - 



+ 1 = 0. 



It follows that the distant points of the curve approach the 

parabola 

2X° + y = O (3) 

This parabola is said to give the form at infinity * of a branch 
of the curve. Since, in equation (3), x may have either sign, 
but y must be negative, the infinite branches in question lie 
in the third and fourth quadrants. These branches recede 
indefinitely from both axes, but they tend to parallelism to 
the axis of y. 

Case in which one of the Axes is an Asymptote, 

273. In the case of the point at infinity in the direction 
of the axis of x, indicated by the factor y in the highest term 

* In general, the parabola thus found, while it gives the form at infinity, is not 
the asymptotic parabola, or that to which the curve approaches indefinitely The 
equation of that parabola is found by evaluating the expression in the first member 
(in this case 2x 2 -\-y) which takes the form 00 — 00 , exactly as in the process for the 
rectilinear asymptote, Art. 264. In the present case, however, the parabola (3) is 
asymptotic, as will be seen by multiplying equation (2) by y and then making y 
infinite, 



XXV.] ONE OF THE AXES AN ASYMPTOTE. 



271 



of equation (1), Art. 272, we have, by equating to zero the 
coefficient of x 2 (see Art. 266), the equation y = o, indicating 
that the axis is itself the asymptote. When this is the case, 
it is easy to ascertain on which side of the asymptote the 
curve lies at either end. For, since y tends to the limit zero 
as x becomes infinite, xy or some other product of powers will 
tend to a finite limit when x is infinite. Now, dividing equa- 
tion (1) by x, we have 

y 3 

2x y+x +4 = x ; 

and, for the point at infinity, this reduces to 

xy -f- 2 = o. 

It follows that, for the distant points of the curve, x and y 
have opposite signs. Thus the branch approaching the right 
end lies below the axis, and that approaching the left end lies 
above it. 

274. The curve 

2x 2 y + y 1 +4*= 3. (0 

whose infinite branches are considered in Arts. 272 and 
273, is traced in Fig. 42. It 
intersects the axis of y in the two 
points (o, ± 4/3), A and B in the 
figure ; and the axis of x in the 
single point C, (f , o). The con- 
tinuity of the branches now re- 
quires us to join the infinite 
branch in the second quadrant 
to A and then to C, and that in 
the third quadrant to B; but we Fig. 42. 




272 APPLICATION TO PLANE CURVES [Art. 274. 

are left in doubt as to whether the other infinite branches 
should be joined to one another, or to B and C respectively. 

275. To decide this point, we may search for maxima and 
minima coordinates ; for a maximum and a minimum abscissa 
will exist if the first of these alternatives is the actual mode in 
which the branches join; whereas, in the other case, these 
will be replaced by a maximum and a minimum ordinate. 

Differentiating equation (1) we have 

dy — 2{xy -j- 1) u 
dx~ y -\-x* " v " 

To find horizontal points on the curve (for maximum ordi- 
nate) we must, as in Art. 127, combine the equation w — o 
with that of the curve. Eliminating x from equation (1) by 
means of the equation xy -f- 1 = o, we have the cubic 

y»— 3y— 2 = 0, or (y + if(y- 2) = 0. 

To the root v=2 corresponds x = — J, giving the point 

D in the figure, at which there is a maximum ordinate; but 

the double root y = — 1 gives the point (1, — 1) which makes 

dy o 

v = o also, so that -7- takes the form — . This is, in fact, the 

dx o 

example given in Art. 171 of that case, which indicates a 

double point ; and the gradients of the two branches were 

there found to be — 2 ± 4/6, or — 4.45 and -f- O.45, as shown 

at E in the diagram. 

Examples XXV. 

Find the asymptotes of the following curves : 
1. (x-\- a)y 2 = (jy -f- b)x 2 . 

x = — a,y = — &,y= x -\- h — a 



§ XXV.] EXAMPLES. 273 

2 . x 3 — /\.xy 2 — $x 2 -J- 1 2xy — 1 2y 2 -f- Sx -J- zy + 4 = o. 

a: = — 3, .%• = 2y, a; -f- 2^ = 6. 

3. (y — 2x) (y 2 — x 2 ) — a (y — x) 2 -j- \a\x -\-y) — a 3 = o. 

j/ = x, y -\- x = \a,y = 2X-\-^a. 

4. ■A; 2 / 2 -f- (2:r (.r -j-j^) 2 — 2tf 2 >' 2 — tf 4 = o. x = — 2<2, .%• = tf. 

5. a: 7 — x^y* -j- tf 4 ^ 3 — ax 2 y^ = o. 

j; = o, a: = — #, jv -j-j/ = J<z, jr —j/ = Ja:. 

6. Jt: 2 (^ — j/) 2 — a 2 (x 2 -\- y 2 ) = 0. jt=±<2, y = x ± a ^/z- 

7. 2Jt: (jt — y) 2 — $ a ( x * — y 1 ) ~f~ 40* — -^O^ 2 — 7^ 3 = °* 

8. Trace the curve y 3 = x 2 {x — a). 

9. Trace the curve x 3 — 2x 2 y — 2X 2 — 8y = o, and show that 
( — 2, — i)isa point of inflexion. 

10. Trace the curve y 2 {x — a) = x 2 (x -{- a), and show that the 
origin is an isolated point or acnode. 

11. Trace the curve x 3 -{- y 3 — ^axy = o, which is known as the 
Folium of Descartes. 

12. Trace the curve 2x 2 y — x 2 -\-y 2 -j- 2x = o. 

13. Trace the curve x 3 — y 3 — x 2 — 2xy = o. 

14. Trace the curve x 3 -\- 2X 2 y -\- xy 2 -\- a 2 y =o, showing that it 
has parallel asymptotes. 

15. Trace the curve (x 2 — y 2 ) 2 — 4^ 2 + zy = o. 

16. Trace the curve x 3 — y 3 — x 2 -f- 2 3> 2 — o. 

1 7 . Trace the curve x 3 -j- y 3 — x 2 — y 2 — o, and show that it is 
symmetrical to the line x =y. 

18. Putting k in place of the second member of equation (1), 
Art. 274, trace the general form of the curve when k < 3 and when 
k> 3. 

19. Trace the curve x^ — ax 2 y -f- axy 2 -^- \a 2 y 2 = o ; show that 
there can be no negative values of x. The two branches meeting 
at the origin are said to form a ramphoid cusp. 

20. Trace the curve x 5 — 4<zy 4 -f- 2ax 3 y -j- a 2 xy 2 = o. 



274 APPLICATION TO PLANE CURVES. [Art. 276. 



XXVI. 

Coordinates Expressed in Terms of a Third Variable. 

276. The form of the rectangular equation of a curve 
sometimes suggests the expression of each of the coordinates 
x and y as explicit functions of a third or auxiliary variable. 
For example, the equation of the ellipse, 

x* , y* 



suggests the employment of an auxiliary variable ^, such that 

x* y* 

— - = cos 2 t/>> whence 70- = sin 2 ^. Hence we may put 

a 4 

x = a cos rp, y = b sin ip (2) 

Equations (2) have the advantage of expressing x and y 
as one-valued functions of ip, so that each value of ty distin- 
guishes without ambiguity a single point of the curve. We 
may regard the point (x, y) as describing the whole curve, 
while rp varies from o to 2tt. On the other hand, when x is 
taken as the independent variable, y is a two-valued function, 
while x varies from — a to -)- a. 

211. We may now express the equation of a tangent to 
the ellipse in terms of the tp of the point of contact. Thus, 
differentiating equations (2), 

dx = — a sin t/>dtp, dy = b cos tpdif>, 



§ XXVI.] EMPLOYMENT OF A THIRD VARIABLE. 275 



whence 



tan <p = -^ = 

ax 



<2 



cot ip. 



Now, substituting their values in terms of tp for x x and y 1 in 
equation (1), Art. 246, we have, after reduction, 

ay sin tp -f- bx cos tp = ab 

for the equation of the tangent to the ellipse at the point 
(a cos if>, b sin ip). 

In like manner, for the normal at the same point, we have 



a 
y — b sin tp = -r-tan ip(x — a cos tp) ; 



that is, 



by cos tp — ax sin ip -\- (a 2 — b 2 ) sin ^> cos tp = o. 



Zl^^ Cycloid. 

278. In the case of a number of important curves treated 
of in the following articles, the auxiliary variable is suggested 
by the definition of the curve as a geometrical locus. For 
example, the path described by a 
point in the circumference of a circle 
which rolls upon a straight line is 
called a cycloid. The curve consists 
of an unlimited number of branches 
corresponding to successive revolu- 
tions of the generating circle; a single branch is, however, 
usually termed a cycloid. 




Fig. 43. 



276 APPLICATION TO PLANE CURVES. [Art. 278. 

Let O, Fig. 43, the point where the curve meets the 
straight line, be taken as the origin, let P be the generating 
point of the curve, and denote the angle PCR by tp. Since 
the arc PR is equal to the line OR over which it has rolled, 

OR = PR= aip; 

and, since CM = a cos tf>, PM — a sin tp, we have 

00= a((p — sin tp), | 

y = a(i —cos)ip. ) ' 

In these equations tp = o, gives the coordinates of //jg cws/> of 
the curve situated at the origin, ip = n gives the coordinates 
of the highest point O' or vertex, tp — 2n corresponds to the 
next cusp or extremity of the first branch. 

279. The employment of the two equations (1) is far 
more convenient than that of the single rectangular equation 
which results from eliminating tp between them. For ex- 
ample, let it be required to find the direction of the motion of 
P and its linear velocity when the circle rolls uniformly with 
the angular rate 00. Differentiating equations (1), 

dx = a(i — cos ip)dip, dy = a sin tpdip, 

therefore 

dy sin tp 

tan = — = = cot i tp ; 

ax 1 — cos tp * T 

whence, taking in the direction of the motion when tp in- 
creases, 

0= 90 — %t/> (2) 



§ XXVI.] THE CYCLOID. 277 

Again, squaring and adding we have 

ds 2 = a 2 (2 — 2 cos ip)dip* = 4a 2 sin 2 \ ip dip 2 , 

whence 

ds = 2a sin %ip dip (3) 



ds 
Writing v for the linear velocity -=-, and go for the angular 



dip 

velocity^, 



df 



v = 2dGo sin £i/> = 2V sin \ty> 



where V = aao is the linear velocity of the centre. 

It readily follows from equation (2) that the chord RP of 
the circle is normal to the curve, and from equation (3) that 
the velocity of P in uniform rolling is proportional to RP. 

280. The equations of the cycloid when in the inverted 
position are generally referred to the vertex O' as origin. In 
Fig. 43, O' is the point (an, 2a). Taking this as origin, and 
taking the opposite direction of the axis of y as positive, the 
new coordinates are x' = x — an, y' = 2a— y, therefore. 

%' = a(ip — n — sin *p) and y' = a{\ -f- cos tp) ; 

but, in this case, it is more convenient to put *p f for tp — it\ 

thus 

x' = a(*p' ~f- sin tp r ), 



y' — a(i — cos tp').\ ^ 

In these equations, tp' = o corresponds to the vertex at the 
origin, and tp' = ±n corresponds to the cusps (± an, 2a). 



278 



APPLICATION TO PLANE CURVES. [Art. 28 1. 



The Prolate and Curtate Cycloids. 



Fig. 44. 



281. The curve described by a point in the plane of the 
rolling circle, either within or with- 
out the circle itself, is called a 
trochoid. Denoting by b the dis- 
tance CP of the point from the cen- 
tre, Fig. 44, and using the same 
notation as in Art. 278, so that OR 

is equal to the arc a>p subtending the angle RCP, we have 
Fig. 45. 




a\p R 

Prolate cycloid. 




x = dip — b sin ip, 
y = a — b cos *p. 



(1) 



Curtate cycloid. 



When b<Ca, the curve is the prolate 
cycloid, Fig. 44, and when 6>a, the 
curtate cycloid, Fig. 45. 



The Epicycloid and the Epitrochoid. 

282. When a circle, tangent to a 
fixed circle externally, rolls upon it, 
the path described by a point in the 
circumference of the rolling circle is 
called an epicycloid. 

Taking the origin at the centre 
of the fixed circle, and the axis of x 
passing through A (one of the posi- 
tions of P when in contact with the 
fixed circle), a, b, i/> and X being 
defined by the diagram, we have, evidently, 




Fig. 46. 



arp = bX, 



a 



§ XXVI.] EPICYCLOID AND EPITROCHOID. 



279 



The inclination of PC to the axis of x is equal to tp -J- x-> 
that is to — 7 — ip ; the coordinates of P are found by subtract- 
ing the projections of PC on the axes from the corresponding 
projections of 0C\ hence 

a + b 
x = (a + 0) cos tp — b cos — 7 — ip, 



a • 1 ■ b 
y = (a -f- b) sin tp — b sin — 7 — ip. 



■ ■ ■ (1) 



283. If the describing point is taken on the radius CP 
at a distance c from the centre C, the curve described is called 
an epitrochoid. (When c>&, this is a looped curve as in Fig. 
46.) Hence the equations of the epitrochoid are found by 
replacing the projections of b in equations (1) by those of c; 
thus they are 



x = (a -\- 0) cos w — c cos — =- — ip, 

, , n . . a-\- b 

y = [a -|- 0) sin tp — c sin — =■ — ip. 



K • 



(2) 



In equations (1), the axis of x passes through a cusp, and in 
equations (2), through one of the points nearest to the origin. 
If we change the sign of c, we have 



x = (a -f- b) cos ip -[~ c cos — =- — ip; 



y = (a -f- b) sin tp -\- c sin 



a+6 



0, 



(3) 



for the curve described by a point on the radius PC produced 
through C. Thus equations (3) are those of the epitrochoid 



28o 



APPLICATION TO PLANE CURVES. [Art. 283. 



when ^vertex, or one of the points farthest from 0, is situated 
upon the axis of x. If c = b, they become the equations of 
the epicycloid under the same circumstances. 




The Hypocycloid and the Hypotrochoid. 

284-. When the rolling circle has 

internal contact with the fixed circle, 

the curve generated by a point on the 

circumference is called a hypocycloid, 

whether the radius of the rolling circle 

be greater or less than that of the 

fixed circle. Curves generated by 

FlG - 47- points on the radius, either within or 

without the circumference of the rolling circle, are called 

hypotroc holds . 

Adopting the notation used in deducing the equations of 
the epitrochoid, we have (see Fig. 47) 

OC = a — b, and X — rf. 

The inclination of CP to the negative direction of the axis of 
x (an acute angle in the diagram) is 



* = 



a 



■t\ 



hence the equations of the hypocycloid are 

a - 



x = (a — b) cos tp -f- b cos 
y = (a — b) sin ^ — b sin 



a— b 



(4) 



§ XXVI.] HYPOCYCLOTD AND HYPOTROCHOID. 28 1 



In like manner, the equations of the hypotrochoid described 
by the point P' at a distance c from the centre are 



x = (a - 


a — b 

- 0) COS *p -f- C COS 7 r> 


y — {a - 


. . a — b 


U J bill *p C bill * W • 



• • • (5) 



285. The equations of the epicycloid become those of 
the hypocycloid by changing the sign of b. Compare equa- 
tions (i) and (4). So also equations (3) become equations 
(5) without changing the sign of c; because, in equations (5) 
as well as in equations (3), ip = O gives one of the points 
farthest from the origin (see the dotted curve in Fig. 47). 

These curves may all be included under a common defini- 
tion or mode of generation. For, in Figs. 46 and 47, the 
point C describes a circle whose radius is R = a -\- b, b being 
negative in Fig. 47. At the same time, P describes a circle 
whose radius is c, about the moving point C as a centre. 
The rates of rotation of the radii R and c have a constant 
ratio, but in Fig. 47 the directions are opposite. Now 
putting in equations (3), Art. 283, 



a -f- = R, — 7 — = nz, 



we have 



x = R cos tp + c cos mjp y ) 
y = R sin tp -f- c sin mtp t ) 

in which m is negative when b is negative and numerically 
less than a, as is the case in Fig. 47. 

In this point of view, the curves are called epicyclics. 



282 APPLICATION TO PLANE CURVES. [Art. 286. 

286. It will be noticed that c and R in equations (6) may 
be interchanged, so that c becomes the radius of the fixed 
circle and R that of the one with moving centre, the relative 
rate of rotation being now i/m. This may be shown geo- 
metrically by means of a jointed parallelogram OCPQ, of 
which the sides OC and OQ (of lengths R and c) revolve 
about the fixed point O with rates of rotation having a con- 
stant ratio.* The opposite sides, being parallel to OC and 
OQ respectively, revolve at the same rates about the moving 
centres. Thus P describes the epicyclic, and the order in 
which R and c are taken is immaterial. 

287. The relations in Art. 285 between the constants 
which occur in the form (3) and in the form (6) give 

b = — , a = ~ R, c = c, . . (1) 

mm v J 

for the reduction from the latter form to the form (3). 

From what is shown in the preceding article it follows 
that, when a curve is given as an epitrochoid or hypotrochoid, 
there is a second method of generating it as such. For, 
after the equation is reduced to the form (6), we may inter- 
change R and c (changing m to i/m), that is, put 

R' = c, c' = R, m' — - , 

m 

and then find constants a' , V and d for a new expression in 
the form (3) by means of equations identical in form with 
equations (1). The resulting values will be found to be 

a' = -^, ¥ =±±^, C '= a + b . . ( 2 ) 

* See Fig. 48, in which the initial position of OC and OQ is in the axis of x 
and the angular rate of OQ is three times that of OC. 



§ XXVL] epjcyclics. 283 

288 The following relation between the radii a, b, a 1 
and b' is noteworthy. The equations above give 

V a + b b' , b 

— — — , or - H = — I . 

a a a a 

This shows that when a and b have the same sign (as in 
the epitrochoid), b' is opposite in sign to a', and is numerically 
the greater. Therefore the epitrochoid can be generated as 
a hypotrochoid in which the radius of the rolling is greater 
than that of the fixed circle. 

On the other hand, when b is negative and numerically 
less than a, b' is negative and numerically less than a\ the 
curve is a hypotrochoid with rolling circle smaller than the 
fixed circle in each mode of generation, and the numerical 
sum of the ratios is unity. 

289. As an example of the double mode of generation, 
let us take the epitrochoid in which 

i j 3 

a = 2, 0=1 and c = — 

4 

in the form (3), Art. 283, which we have taken as the stand- 
ard, the positive sign of c indicating that in the initial position 
c is measured away from the origin. The equations are 



3 
x = 3 cos i/j -}- — cos 3^, 
4 

3 

y = 3 sin tp + - sin 3^, 

so that, in the notation of Art. 285, 

3 
£= 3> c = -, m=3. 
4 



(0 



284 



APPLICATION TO PLANE CURVES. [Art. 289. 



Now equations (2), Art. 287, give 



a' = — U, 



&' = 2j, c' = 3, 



for the constants in the second mode of generation, in which 

the curve is (in accordance with 
Art. 288) a hypotrochoid. 

In Fig. 48, the initial positions 
of the rolling circles (corresponding 
to ip = o) in each mode of genera- 
tion are shown, together with the 
position of the parallelogram of Art. 
286 for a value of ip about 25 . 
IG ' 4 This particular curve is symmet- 

rical to both axes, because the point B nearest to the origin 
falls upon the axis of y. 




Algebraic Forms of the Equations. 



290. When the radii of the rolling and fixed circles are 
commensurable, the points of the circumferences which were 
originally in contact will again be in contact after a certain 
number of revolutions. In this case, the curve will begin to 
repeat itself, so that it consists of a finite number of branches. 
It will then admit of an algebraic equation, which is the result 
of eliminating ip from its two equations. 

For example, if a = 2b, the equations of the hypotrochoid, 
(5), Art. 284, become 



x = (b -[- c) cos tp 9 \ 
y = (b — c) sin tp, j 



. . (1) 



§ XXVI.] ALGEBRAIC FORMS OF THE EQUATIONS. 285 

which, see Art. 276, are the equations of the ellipse 

x 2 f 
{b + cf + [b - cf =l ( 2 } 

Thus the hypotrochoid becomes an ellipse when the rolling 
circle is one-half the size of the fixed circle. Putting c = b 
in equations (1), we have y = O, showing that the correspond- 
ing hypocycloid is a straight line ; that is to say, every point 
in the circumference of the rolling circle describes a diameter 
of the fixed one. 

291. In the cycloidal cases, in which c = b, the same fixed 
circle serves in each of the modes of generation, and the cusps 
are situated upon it. In the cases now under consideration 
(the radii being commensurable), the cusps divide the circum- 
ference into equal parts. The epicycloids and hypocycloids 
may, in these cases, be distinguished by specifying the number 
of cusps, say r, together with, if necessary, the number of 
rth parts of the circumference covered by one branch. If 
more than one segment is thus covered, the branches cross 
one another. 

In the case of the proper hypocycloids, which lie within 
the fixed circle, the sum of the radii of the rolling circles in 
the two modes of generation must, by Art. 288, be equal 
to a. Thus there is but one three-cusped hypocycloid, and 
in it the value of b is either -J-a or \a\ but there are two five- 
cusped hypocycloids, one of which is generated when b = \a 
or fa, and is uncrossed, while the other, which is crossed or 
has double points, is generated when b = \a or fa. 

Again, if b = a, we have a one-cusped epicycloid which 
simply surrounds the fixed circle. (This curve is known as 
the cardioid and will be discussed later under another defini- 



286 



APPLICATION TO PLANE CURVES. [Art. 29 1.. 



tion.) But if b = 2a, we have an epicycloid with a single 
cusp, one branch of which enwraps the fixed circle twice. 

The Four-cusped Hypocycloid or Astroid. 

292. The four-cusped hypocycloid may be generated by 

a rolling circle whose radius is \ that 
of the fixed circle, as indicated in Fig. 
49, or by one whose radius is j of <z. 
Putting b = Ja in equations (4), Art. 
284, we have 

x = — acos tp -f- —a cos 3^, 

Fig. 49. y = -a sin tp — -a sin 3^. 

By means of the formulae 

cos 30 = 4 cos s ip — 3 cos tp t 
sin 3?/> = 3 sin tp — 4 sin 3 ^, 

these reduce to 

X=(I cos 3 *p, 
y = a s'm 3 ip. 

Eliminating tp, we have 




x* -\- y 



a 



an equation which, when freed from radicals, is found to be of 
the sixth degree. This curve is sometimes called the Astroid. 

Employment of m as an Auxiliary Variable. 

293. When a curve whose rectangular equation is given 
has a multiple point at the origin, it is frequently convenient 



§ XXVI.] m AS AN AUXILIARY VARIABLE. 287 

to express x and y in terms of m, where y = mx. If the curve 
is of the wth degree and has r branches passing through the 
origin, the straight line whose equation is y = mx cuts the 
curve r times at the origin, and can therefore cut it in no more 
than n — r other points. In fact, if we substitute y = mx in 
the equation, every term in the result will contain x r or a 
higher power of x. Dividing by x r we then have an equation 
of the (n — r)th degree for x in terms of m. 
294. For example, given the curve 

x* — $axy 2 -(- 2ay z = o, 

which is of the fourth degree, and has a triple point at the 
origin. Putting y = mx, we have 

# 4 — (3#w 2 — 2am 3 )x 3 = o, 

and, rejecting the factor x s which gives three roots equal to 
zero, 



whence 



x = 3am 2 — 2am s ; 



y = 3am 3 — 2 am 4 . 



Thus x and y are, in this case, one-valued functions of m; 
and, by giving particular values to m, we may determine as 
many points as we please upon the curve. The 
values of m which make x and y vanish deter- 
mine the tangents at the origin. They are, in 
this case, m= o and m = J-. It is clear that, 
as m increases from o to J-, the line y = mx 
turns about the origin, and the point (x, y) upon 
it describes a loop of the curve returning to FlG - 5°- 
the origin, see Fig. 50. As m increases from J- to 00 , x and y 




288 APPLICATION TO PLANE CURVES. [Art. 294. 



pass from o to- 00 the point (x, y) describing the branch 
in the third quadrant. Finally, as m changes sign and passes 
from — 00 to o, x becomes positive and y negative, and the 
point (x, y) describes the branch in the fourth quadrant 
returning to the origin. 

295. The maximum values of x and y occurring in the 
loop may be determined from their expressions in terms of 
m. Equating to zero the derivative of x, we thus find m = 1 
and m = o ; the former gives the point (a, a) (A in the dia- 
gram) at which x is a maximum, and the latter the origin at 
which x in the cusp is a minimum of the variety shown in 
Fig. 27, Art. 133. 

In like manner, from the derivative of y, we find m 2 = o 
and m = f ; the former corresponds to a double root for 
which the derivative does not change sign, and the latter 
gives the point of maximum ordinate at about (0.95a, 1.07a). 

These values determine the loop with considerable accu- 
racy. The branch in the fourth quadrant is equally well 
determined by the point B for which m = — J, namely, 
(a, — ia), and the slope at that point, as given by the value 
of dy/dx. 

296. The method is equally applicable when the origin is 
an isolated point or acnode. For example, the symmetrical 
curve 

(x 2 -f- y 2 ) 2 = 4X 2 + y z 



gives 



4 + m 2 



X' = 



( 1 -\-ni 2 ) 



w 



4m 2 -f- m 4 
y = (1 + m 2 ) 2 ' 



Here x and y cannot become zero, nor can they become in- 



§ XXVI.] EXAMPLES. 289 

finite. The value m = o gives the points (± 2, o) on the 
axis of x, and m = 00 gives (o, ± 1) on the axis of y. Treat- 
ing y* as a function of m 2 , it is readily shown that m = ± |/2 
gives points of maximum ordinates at (± -J |/6, ± f 1/3). 

Examples XXVI. 

1. The locus of the point M in Fig. 43 was called by Roberval 
"the companion to the cycloid." Show that it is a curve of sines 
(see Fig 12, p. 63) symmetrical to the centre of the rectangle OO f and 
therefore bisects its area. 

The area between the two curves regarded as generated by the vari- 
able line PM parallel to OX is readily perceived to be equal to the area of 
the semicircle, which is generated by this line when the point M describes 
a diameter of the circle regarded as fixed. In this way Roberval proved 
that the area of the cycloid is three times thai of the generating circle. 

2. Prove that in the trochoid, as well as in the cycloid, the line 
PR is a normal and is proportional to the actual velocity of P in 
uniform rolling. 

3. Determine the ordinate of the point of inflexion in the pro- 
late cycloid. a 2 — b 2 

■ y = — ~« — ' 
a 

4. Using the general equations (6), Art. 285, show that points of 
inflexion occur when 

1P+ m 5 c 2 
cos (m — i)ip = ■ — , 

and hence show that the epitrochoid has points of inflexion when the 

b 2 
numerical value of c lies between b and — ■ — ,, and similarly for the 

a -\- b 

hypotrochoid. 

5. Derive the algebraic equation of the two-cusped epicycloid with 
cusps on the axis of x. ^{x 2 -\-y 2 — # 2 ) 3 = 2 7#^ 2 . 



29O APPLICATION TO PLANE CURVES. [Ex. XXVL 

6. Show that in the four-cusped hypocycloid the auxiliary angle 
ip is the inclination of the tangent at (x,y) to the negative direction of 
the axis of x; find also the value of the perpendicular from the origin. 

p = a sin ip cos ip. 

7. Show that in the four-cusped hypocycloid the intercept of the 
tangent between the axes is constant; also that the point of contact 
and the foot of the perpendicular are equally distant ' from opposite 
extremities of this line. 

8. Determine the value offl for the epicycloid. 

p = fa 4- 20) sin — -. 
s 20 

9. Trace the curve y 4 — X* -f- 2axy 2 = o. 

10. Trace the curve x 5 -\-y 5 — $ax 2 y 2 = o. 

11. Trace the curve x 5 -\-y 5 — ^ax z y = o. 

12. Trace the curve x 5 -\-j> 5 — 2a z xy = o. 

13. Trace the curve x° — y % -f- ( 2 _y — ^) 2 — °« 

14. Trace the curve y 4 — g6a 2 y 2 -\- iooa 2 x 2 — x 4 = o, which has 
been called "la courbedu diable." Determine maximum and mini- 
mum values of x. 

15. Trace the curve x 4 -\-y* -\- 6ax 2 y — Say 3 = o. 

16. Trace the curve x r ° -\-y 4 — $x 2 y — 3^ry 2 = o, determining 
points by putting m = 1, and finding the slope at those points. 

17. Trace the curve x l — ax 2 y — axy 2 -f- a 2 y 2 = o, showing that 
the line j' — mx touches the curve when m = 1 and when m = — 3. 
Find a maximum value of x by solving for_y. 

18. Trace the curve x 4 — 2a 2 x 2 — 2<2>' 3 -J- $a 2 y 2 = o, showing that 
it has nodes at points corresponding to m = ± 1. 



XXVII.] 



POLAR EQUATIONS. 



29I 



XXVII. 



Polar Equations. 

297. When the equation of a curve is given in polar 
coordinates, we shall assume that it is solved for r, that is to 
say, given in the form r = /(d). Let s be the length of an 
arc of the curve measured in the direction in which 6 in- 
creases. Then, as 6 increases, the point P in Fig. 51 moves 

ds 
with the velocity -j- . Let PT, a portion of the tangent line, 

represent ds; then, producing r, let the rectangle PT be com- 
pleted, and let ip denote the angle TPS, 
that is, the angle between the positive 
directions of r and 5. The resolved ve- 
locities of P along and perpendicular 

dr . , rdd . 
to the radius vector are -7- and — r— , the 

at at 

latter being the velocity which P would FlG - 5 1 * 

have if r were constant; that is, if P moved in a circle de«* 

scribed with rasa radius. Hence we have 




PS= dr 



and 



PR = rdd. 



From the triangle PST, we derive 



rdd 
tan ib = -7—, 

r dr 



sin ^ = 



rdd 
ds 



cos 



dr 

*=3P • (1) 



292 
and 



APPLICATION TO PLANE CURVES. [Art. 297. 



ds 2 = dr 2 + rW (2) 



In accordance with the assumption that ds has the sign of 
dd, we write 

£=/[-+©•] <» 

and we infer from the second of equations (1) that the 
value of will always be either in the first or in the second 
quadrant. 

The first of equations (1) is equivalent to 



dr 



(4) 



298. It is frequently convenient to employ in place of 
the radius vector its reciprocal, which is usually denoted by 
u; then 



1 _ du 

r = — , and dr = — — «- 

u w 



(5) 



Making these substitutions, equations (3) and (4) give, in 
terms of u and 0, 

%=w\^ + (Tei] • • • • (6) 



and 



du 

cot * = _ USB- 



(7) 



§ XXVI I.] POLAR SUBTANGENTS AND SUBNORMALS. 293 



Polar Subtangents and Stibnormals. 

299. Let a straight line perpendicular to the radius vec- 
tor be drawn through the pole, and let the tangent and the 
normal meet this line in T and N respect- 
ively ; then the projections of PT and ^ 
PN upon this line, that is OT and ON, 
are called respectively the polar subtan- 
gent and the polar subnormal. In Fig. 52, 
OPT — tp ; whence 

dr du' 

and 



OT 



r tan ip = r~r~ = 




dr 
ON =r cot ip — -=-= 



du 
uHQ' 



Fig. 52 shows that the value of OT is positive when its 
direction is 6 — 90 ; that of ON is, on the other hand, posi- 
tive when its direction is 6 -\- 90 . 

The Perpendicular from the Pole upon the Tangent. 

300. Let p denote the perpendicular distance from the 
pole to the tangent; then, from Fig. 52, we obtain 



p — r sin ip =r 2 -=- = 



dd 
ds 



r 2 



/[>+( 



dr \ 2 ~ 
dd 



• (1) 



ds 



These expressions give positive values for p, because — is 

dd 

assumed to be positive, and Fig. 52 shows that p has the 



294 APPLICATION TO PLANE CURVES. [Art. 3OO. 

direction <fi — 90 ; <j) being the angle which the positive 
direction of s makes with the initial line. 
Equation (1) may be written in the form 

. 1 ds 2 



f r'dff* ' 
and, transforming by the formulae of Art. 298, we have 

1 . (du\ 2 

Critical Points. 

301. The critical points of a curve with reference to polar 
coordinates (analogous to the horizontal and vertical points 
of Arts. 127 et seq.) are those for which tf; = o, and those for 
which ip = 90 , while has a finite value. In the first case, 
the radius vector is tangent to the curve. As the moving 
point P describing the curve passes through such a point, dr 
has a finite value while dO — o. Both the polar subtangent 
and the perpendicular p then vanish (see Arts. 299 and 300). 
Unless the point is at the same time a point of inflexion, 6 
has a limiting value such that, as 6 passes through it, two real 
values of r become equal and then imaginary. 

In the second case, when tp = 90 , the radius vector is 
normal to the curve. We now have p = r, while the sub- 
tangent is infinite and the subnormal vanishes. The radius 
vector will, in this case, generally have either a maximum or 
a minimum value. 



XXVII.] ZERO VALUES OF r. 295 



Zero Values of r. 

302. Let r — o when 6 takes the value 6 1 ; then, as 6 
passes through the value 6 1 , the point describing the curve 
reaches the pole moving in the direction of the straight line 
whose inclination is B v Accordingly, the equations of Art. 
297 show that ip = o or 180 , and ds = ± dr. In general, r 
will be found to pass through the value zero and become 
negative as 6 passes through the value 6 1 , so that the curve 
lies as usual upon one side of the tangent line. 

As an illustration, let us take the curve 

r = a cos 6 cos 26 (1) 

Here r = 0. when cos 6 = and when cos 2 = o, that is, 
when 0z=go°, 45 or 135 . The dotted lines in Fig. 53 are 
the tangents at the pole. When = o, 
r = a. Hence the generating point, start- 
ing from A, describes the half loop in the 
first quadrant while 6 increases from O to 
45 . When 6 passes 45 , r becomes nega- FlG - 53- 

tive, as indicated in the diagram, but it returns to zero when 
= 90 , the point (r, 0) describing the loop situated in the 
third quadrant. As 6 passes 90 , r again becomes positive 
and the loop in the second quadrant is described. Finally, 
while 6 passes from 135° to 180 , r is again negative, and the 
point A is reached with the values r = — a, 6 = 180 . 

In this example, the change of d to 6 -\- n, in equation 
(1), changes the sign, but not the value of r; and, since 
(r, 6) and ( — r, 6 -\- n) represent the same point, the curve 
repeats itself when 6 varies from n to 2n, from 2n to 3?r and 
so on. 




296 APPLICATION TO PLANE CURVES. [Art. 303. 

303. The maximum positive and negative values of r will 
be found to correspond to 6 = o and = ± tan -1 5. But the 
form of the loops is better determined, in this case, by the 
maximum value of the ordinate y when the initial line is 
taken as the axis of x. From equation (1), we have 

y = r sin 6 = a sin cos cos 20 = \a sin 46. 

The maximum positive and negative values occur when 
4.O = %7T, Jzr, ^7t, etc., that is when = ■J-n-, f 7T, f 7r and ^7r, 
and the numerical value of the maximum is in each case \a. 

The Lemmscate. 

304. The curve whose polar equation is 

r 2 = a 2 cos 2/9, (1) 

known as the Lemniscate of Bernoulli, will serve to illustrate the 
case in which the value of 6 which makes r = o is also a limit- 
ing value of 6, Art. 301. In equation (1), putting cos 20 = o, 
we have &= 45 , and 0= 135 for the 
^A values which make r 2 = o. When varies 
from o to 45 , r is a two-valued function 
Fig. 54. having numerically equal positive and nega- 

tive values. These values decrease from the initial value ±a 
to zero. Thus the two generating points, starting from A and 
B, Fig. 54, describe the half-loops in the first and third quad- 
rants, and meet at the origin, forming a point of inflexion. As 
passes through 45 , the values of r become imaginary, and 
so remain for all values between 45 and 1 35 ; the rest of 
the curve being described when varies from 135 to 180 . 




§ XXVII.] THE LEMNISCATE. 297 

Or we may regard the whole curve as described while 6 varies 
continuously from — \n to + \n, the whole right-hand loop 
then corresponding to positive values of r. 

305. In finding the maximum ordinates, we may put the 
function y 2 = r 2 s'm 2 6 a maximum. Thus, from equation (1), 

sin 2 6 cos 26 = a maximum ; 
whence 

2 sin 6 cos 6 cos 26—2 s\i\ 2 6 sin 2d = o, 
or 

sin 6 cos 3# == o. 

The root sin 6 = makes y 2 (but not y) a minimum ; but 
cos 3$ = o gives 6 = ± 30 for the vectorial angles of the max- 
imum ordinates. The corresponding values of y are ± \a j/2. 



Polar Equations involving only Trigonometric 
Functions of 0. 

306. When, in the polar equation r =f(6), only trigono- 
metric functions of angles commensurable with 6 occur, r will 
be a periodic function of 6 and the period will be commen- 
surable with 2n. Thus the period of r = a cos %d is 47c be- 
cause adding 4JT to 6 is equivalent to adding 27t to \B. In 
other words, the radius vector returns at 6 = 4ft, for the first 
time, to its initial value, a. Since this period 4?r is a multiple 
of 27t, the generating point returns to its initial position, and 
the curve is completed when the vectorial angle has made two 
complete revolutions. 

Again, the period of r = a cos 26 is tt, because adding 
n to 6 adds 27T to 26, that is r returns to its initial value a 



298 APPLICATION TO PLANE CURVES. [Art. 306. 

when 6 = 7t . But the generating point does not return to its 
original position until 6 = 2n when the curve is completed. 

In each of these examples, the student should trace the 
curve, noting how the generating point completes its circuit 
with alternately positive and negative values of r. In each 
case, the loops so formed are similar; and in the second case 
(there being two corresponding to values of each sign, because 
the period of r is n), the curve consists of four equal loops. 

307. In general, the curve is completed with n revolutions 
of the vectorial angle, where n is the least integer which makes 
2nn a multiple of the period of V. But, when n is an odd 
number, it may happen that the curve will be completed when 
B=.mz\ namely, when the addition of nn to 6 changes the 
sign but not the numerical value of r. We have had an ex- 
ample in the curve of Art. 302, in which the period of r is 27T, 
but the addition of n to changes r to — r, hence the curve 
is completed when = n. 

Again, in r = a cos 3$, the period of r is |7r, and 2n is a 
multiple of this period. But the addition of 7t to 6 changes the 
sign of r, hence the curve is completed with three equal loops 
when 6 = n\ and, when 6 varies from n to 2zr, these loops are 
repeated with values of r opposite in sign. 

Again, for the curve r = a cos id the period of r is 6it, so 
that n = 3 ; but, for the same reason as in the preceding exam- 
ples, the curve is completed when 6 = 3^. 



The Limagon of Pascal. 

308. The curve obtained by producing the radius vector 
of a given curve by a fixed amount is called a protraction of 
the given curve with respect to the pole. The protraction of 



§ XXVII.] 



THE LI MA CON OF PASCAL. 



2 99 



a circle with respect to a point on its circumference is a curve 
named by Pascal the Limagon. Taking the diameter through 
the given point on the circumference, Fig. 55, as the initial 
line, and denoting the radius O'Q by a, the polar equation of 
the circle whose radius is a is 

OQ = 2a cos 6. . . (1) 

Hence, if b is the constant amount of 
protraction, 

r = 2a cos 6 -\- b . . (2) 

is the equation of the limacon. 
The equation 

r = 2a cos 6 — b . . (3) 
represents the same curve. For, if in 
equation (2) we add n to 6, we obtain 





/ 








\ 
\ 








sQ 




\ P ' 
\ 1 

\ 1 


'\ 






M 


"C 


\ / 
A / 


°/ 




' 0' 






1 / 
/ / 
// 
/ 




x: 


N. 


— 


sy 





Fig. 55. 



r — — 2a cos -|- & = — (2a cos # — 6), 

the negative of the value corresponding to 6 in equation (1); 
thus 6 + n in equation (2) gives the same point that gives 
in equation (3). 

309. If we join Q, the moving point on the circumference 
of the circle, with its centre O', the angle QO'A is double the 
angle QOA or 0. Hence it is evident that the limagon may 
be generated by the epicyclic method of Art. 285. The 
radius O'Q here revolves with double the angular rate of QP. 
Thus the equations of the curve, as referred to the origin O', 
may be written in the form 



x = b cos 6 -f- a cos 26, 
y=bs'm 6 -f- a sin 2d, 



300 APPLICATION TO PLANE CURVES. [Art. 309. 

in which, comparing with equations (6), Art. 285, 

R = b y c =a, m — 2. 

Since m is positive, the curve is an epitrochoid, and denoting 
the values of its constants when written in the form (3), 
Art. 283, by a', b f and c', they are, by equations (1), Art. 
287, 

a' = %b y &' = %b, c' = a. 

Thus the limacon is an epitrochoid in which the fixed and 
rolling circles are equal. 

310. In accordance with Art. 288, the limagon may also 
be generated as a hypotrochoid, the constants for this mode 
being, by equations (2), Art. 287, 

a" = — a, b"= 2a, c" = b. 

Thus the fixed circle is, in this case, identical with the pro- 
tracted circle drawn in Fig. 55, and the rolling circle has a 
radius equal to the diameter of that circle. The initial posi- 
tion of the point of contact is at O, and the centre of the 
rolling circle lies always on the circumference of the fixed one ; 
for example, it is at Q in Fig. 55 when the generating point 
is at P. 

The limagon in which b>2a does not pass through the 
pole O, and therefore forms a single oval as in Fig. 56. 

The Dygogram. 

311. The limagon occurs in the graphic representation of 
magnetic forces introduced by Archibald Smith, Esq.,* in 

* Transactions of the Institution of Naval Architects, vol iii, p. 70. 



§ XXVIL] 



THE DYGOGRAM. 



3d 



1862, into the Theory of the Deviation of the Compass in 
iron ships. By the principles of Mechanics, the several forces 
are represented by lines having proper directions and magni- 
tudes, and their joint effect is obtained by laying them off 
successively, end from end, beginning at a fixed point. The 
forces which vary in direction relatively to the meridian for 
different positions of the ship's head are two in number, 
known respectively as the semicircular and the quadrantal de- 
viating forces. They are constant in magnitude ; and, as 
the ship swings completely round, 
the former makes one revolution in 
direction and the latter two. Hence, 
when laid off from O' (the end of a 
line representing the constant force), 
they take just such positions as O'Q 
and QP in Figs. 55 and 56, which 
revolve one at twice the angular rate 
of the other, and the final point P 
describes a limacon as explained in 
Art. 309. 

The limagon thus used, with points corresponding to 
different headings of the ship marked upon its perimeter, 
is called the Dygograrn,* and serves to determine the direction 
of the total magnetic force for every position of the ship's 
head. 




Fig. 56. 



* A contraction of dynamo — gonio — gram from SvvajuiS, force, and ■yoaria, 
angle. Mr. Smith in his diagram laid off from the fixed point the slowly rotating 
line representing the semicircular force (which is usually the larger, as O'C in 
Fig. 56). The locus of its extremity is the dotted circle. From points of this 
circle the lines representing the quadrantal force were then drawn at inclinations 
in each case double that of (7 C, as in the construction of the epicycle. The oppo- 
site order of construction, as in DiehVs Compensation of the Compass, p. 31, cor- 
responds more directly to the definition of the curve as a limacon. 




302 APPLICATION TO PLANE CURVES. [Art. 3 1 2. 

The Cardioid. 

312. When b= 2a, the limagon becomes a cusped curve, 
which, by Art. 309, is the epicycloid formed when the fixed and 

the rolling circles are equal. In Fig. 57 
the cardioid is drawn with its cusp on the 
right. Denoting the diameter of the fixed 
circle by a, the polar equation of the curve 
in this position is 

r = a(i — cos 6) = 2a sin 2 \6. . (i) 

If the tangent common to the fixed and the 

rolling circle at their point of contact be 

drawn, the equality of the arcs OT and PT shows that the 

radius vector OP is bisected at right angles by the tangent 

at Q. It follows that the locus of Q is 

r = a sin 2 %v f (2) 

a cardioid of one-half the linear dimension of the locus of P. 
The locus of the foot of the perpendicular from a fixed point 
upon a tangent to a given curve is called a pedal of the given 
curve. Hence it follows that the cardioid (2), represented 
by the dotted line in the figure, is the pedal of the circle whose 
diameter is a, with respect to a point on its circumference. 



Transformation to Rectangular Coordinates. 

313. The equations of transformation from rectangular to 
polar coordinates, the origin being the pole and the axis of % 
the initial line, are 

x = r cos 0, y = r sin 6. 



§ XXVII.] INFINITE VALUES OF r. 303 

In the reverse transformation the trigonometric functions 
are expressed in terms of sin 6 and cos 6. Then, substituting 
y/r and x/r respectively for these, and clearing of fractions, r 
is finally eliminated by means of the relation 

r 2 = x 2 -J- y 2 . 

For example, in the case of the limacon, equation (2), Art. 

308, becomes 

x 
r = 2a— + b: 
r 

whence 

x 2 -j- y 2 = 2ax -f b ^/(x 2 -f- y 2 ) , 

and clearing of radicals, we have 

(#2 -}- y 2 ) 2 — 4 ax(x 2 + f) + (4a 2 — b 2 )x 2 — by = o 

for the rectangular equation. 

The equation indicates a double point at the origin ; but, 
when b > 4a, the tangents at the origin become imaginary and 
the origin appears as an isolated point, the curve then having 
the form given in Fig. 56. 

The rectangular equation of the cardioid, equation (l), 
Art. 313, is 

(x 2 + y 2 ) 2 -f- 2a(x 2 -{- y 2 ) — a?y 2 = o. 

Infinite Values of r. 

314. When r becomes infinite for a finite value of 6, the 
curve r=f(d) has a point at infinity in the direction indicated 
by this value of 6. Compare Art. 261, in which the direction 



304 APPLICATION TO PLANE CURVES. [Art. 3 1 4. 

y 

ratio m is the value of the ratio — or tan 0, when x, y and r 

become infinite. 

The infinite branch of the curve will have an asymptote, if 
the perpendicular upon the tangent has a finite limit when 
the point of tangency recedes to infinity. It is obvious from 
Fig. 52, p. 293, that when P is at infinity the subtangent OT 
coincides with p. Hence if d x is the value of 6 which makes 
r infinite, the perpendicular upon the asymptote will be 

, the former being laid off in the direction 

flj — 90 when positive, and the latter in the 
direction 6 -f- 90 when positive. 

315. The expression deduced below is some- 
times more convenient. In Fig. 58 the line 
OB is drawn through the pole in the direction 
V Then, dropping the perpendicular PB from 
a point P of the curve upon this line, we have, 
from the triangle OPB, 




du 



Fig. 58. 



PB = r sin (6 1 - 6). 

Now, if the curve has an asymptote parallel to OB, it is 
plain that, as 6 approaches 6 V the limiting value of PB will be 
equal to OR, the perpendicular from the pole upon the asymp- 
tote. Hence, if the curve has an asymptote in the direction 
8 V the expression 

OR = [r sin (6 1 - Q)\, 

which takes the form 00 -o, will have a finite value, and this 
value will determine the distance of the asymptote from the 
pole. Fig. 58 shows that when the above expression is posi- 
tive OR is to be laid off in the direction B t — - 90 . 



§ XXVII.] POINTS OF INFLEXION. 305 

If, upon evaluation, the expression for OR is found to be 
infinite, we infer that the infinite branch of the curve is para- 
bolic. 

316. In the special cases, when 6 1 = o and when 6 X = 90 , 
the expression for PB becomes — r sin 6 and r cos 6, respect- 
ively. In rectangular coordinates (the axis of x being the 
initial line, and origin the pole), these are — y and x respect- 
ively. In fact it is obvious that in these cases, namely when 
the asymptote is parallel to one of the axes, the perpendicular 
from the origin is either the value of y when x is infinite, or 
the value of x when y is infinite. 

For example, in the polar equation 

r = a (sec 6 -f- tan 6), 

r is infinite when 6 = 90 ; hence we take 

x = r cos 6 = a(i -f~ sin 6). 

The value of this when d = 90 is 2a; there is therefore an 
asymptote perpendicular to the initial line at the distance 2a 
from the pole. In other words x = 2a is the rectangular 
equation of the asymptote. 

The curve in this illustration is the Strophoid, Fig. 40, 
p. 257, referred to the vertex of the loop as pole. 

Points of Inflexion. 

317. When, as in Fig. 52, the curve lies between the tan- 
gent and the pole, it is obvious that r and p will increase and 

dp 
decrease together ; that is, — will be positive. When, on the 

ar 

other hand, the curve lies on the other side of the tangent, 



306 APPLICATION TO PLANE CURVES. [Art. 3 1 7. 

~ will be negative. Hence at a point of inflexion -f- must 
dr dr 

change sign. It follows that, except at the pole, a point of 

inflexion can occur only where — = O. 

dr 

318. This criterion can be put in a form more convenient 

in application as follows : Taking the derivative of equation 

(2), Art. 300, with respect to u, 

2 dp dud 2 udd 

~~ ~fdu = 2U+2 dMd¥ 2 d^ ; 



hence 



and since 



(■ 



d 2 u\ J_(fy 






this may be written in the form 

d 2 u r 2 dp 
U+ dd i = f'dr'' 

Now, since p is always positive, it follows that the sign of 

~ is the same as that of 
dr 

d 2 u , . 

^ + ^r; ....... (1) 

hence at a point of inflexion this expression must change sign. 



§ XXVII.] 



SPIRALS. 



307 



Spirals. 

319. When f(9) in the polar equation r ---/(d) is not a 
periodic function, the curve is not completed when 6 passes 
over a limited range of values, and it becomes necessary to 
consider all values of 6 from -J- 00 to — 00. Curves of this 
kind are called Spirals. Successive revolutions of the radius 
vector give an unlimited number of successive portions or 
whorls of the spiral. 

Let us first suppose that the infinite value of 6 gives a 
finite value of r. In this case there will be an asymptotic 
circle ; that is to say a circle to whose circumference the suc- 
cessive whorls of the spiral approach indefinitely. 

320. As an illustration, let us take the equation 



ad 2 



r = 



& z - 1 



• (1) 




Equal positive and negative values of 6 give the 
same value of r; hence the curve is symmetrical 
to the initial line. When increases without 
limit, r approaches the limiting value a, the point Fig. 59. 
(r, 6) describes an infinite number of whorls approaching with- 
out limit to the asymptotic circle r = a drawn in Fig. 59. 
When varies from o and 1 , r is negative and varies from o to 
infinity, (r, 6) describing the branch in the third quadrant. 
Evaluating the expression for the perpendicular upon the 
asymptote, Art. 315, we have 



[rsinft-fln = 



- ad 2 sin (1 — 0)-| 
J J 1 6 - 1 



— 1 



^ a. 



The angle 6 = 1 is the radian corresponding to 57 18', 
nearly, and, since the expression for the perpendicular on the 



308 APPLICATION TO PLANE CURVES. [Art. 320. 

asymptote is negative, its direction is ^+90°= 147 18'; 
consequently, the position of the asymptote is that given in 
Fig. 59- When 0>i, r becomes positive and rapidly de- 
creases to approach the limiting value a. 

321. It is obvious that the branch thus described contains 
a point of inflexion. To determine its position we employ the 
criterion of Art. 318. Thus, equation (1) is equivalent to 



u 



a x * 



u du 2 3 - <Pu 6 n , 

whence — - = — , and - — — — -#~ 4 ; 

dd a dd 2 a 



therefore u + ^ = -(i — 6~ 2 - 66-' ) 

dd i a \ / 

y'-d 2 - 6 
~ ad* 

Putting this expression equal to zero, the real roots are 

= ± 4/3, 

and it is evident that, as 6 passes through either of these values, 

d 2 u 
the expression u-\-— changes sign. Hence the points of 

dd 

inflexion are determined by 

D = ± 4/3 and r = — . ' 
The angle = \/?> corresponds to nearly ioo°. 



§ XXVII.] THE SPIRAL OF ARCHIMEDES, 309 



The Spiral of Archimedes. 

322. When the infinite value of 6 makes r infinite, the 
spiral has an infinite number of whorls of increasing magnitude. 
An example is furnished by the simplest of all spirals, that of 
Archimedes , in which the radius vector is 
proportional to the angular coordinate, so 
that the equation is 

r=ad. .... (1) 
The distance between successive whorls is 
constant and equal to 2na.' Since r = o 
when 6 = o, the curve touches the initial 
lines at the pole, as in Fig. 60. 

The equation r = ad -f- b represents the same curve in a 
different position, for we may write it in the form r == a(Q -f- O ), 
showing that the curve differs only in the direction with which 
it reaches the pole. This spiral is, in fact, the only curve 
which is identical with its protractions. 

Negative values of r in equation (1) give a similar spiral 
symmetrically situated to a perpendicular to the initial line. 




Fig. 60. 



The Reciprocal Spiral. 

323. When the infinite value of r makes r = o, the spiral 
has an infinite number of whorls approaching the pole but 
never reaching it. The Reciprocal Spiral whose equation is 

a 

r= j 

_a furnishes an example. See Fig. 61. The value 
of a in the diagram is that of r when 6 is the 
Fig. 61. radian about 57 . 3. When 6 = o, r = 00. 

Hence there is a point at infinity in the direction of the initial 




310 APPLICATION TO PLANE CURVES. [Art. 323. 

line; and, proceeding as in Art. 316, we have 

a sin 6 



y = rsin 6 = 



6 



Evaluating this fraction for 6 = o, we have y = a; hence the 

line 

y = a 
is an asymptote. 

The Logarithmic or Equiangular Spiral. 
324-. The spiral whose equation is 

r = ae nd (1) 




log r = log a + nd . . . (2) 

is called the Logarithmic Spiral. Supposing 

n to be positive, r = 00 when = -|- 00 , and 

Fig. 62. r — o when 6 '= — 00. The curve therefore 

consists of an infinite number of increasing whorls when 6 is 

positive, and it approaches the pole with an infinite number of 

decreasing whorls when d is negative. See Fig. 62. 

Substituting in equation (4), Art. 297, we find for this 
curve 

cot ip. = n; 

therefore the curve makes a constant angle with its radius 
vector. For this reason, it is also called the Equiangular 
Spiral. * 

* From the property of the stereographic projection of the sphere, that angles 
are unchanged in magnitude, it follows that the stereographic projection upon the 
equator of the Loxodromic curve (which cuts the meridians at a constant angle) is 
an equiangular spiral. 



§ XXVII.] THE EQUIANGULAR SPIRAL. 311 

It is readily seen that successive whorls cut any radius 
vector in points whose distances from the pole are in geomet- 
rical progression. Again, the equation r = be ne represents the 
same spiral, only so turned about the pole that the radius vec- 
tor whose length is b coincides with the initial line. Thus 
the logarithmic spiral is identical with all its similar curves. 

Examples XXVII. 

1. Prove that, in the case of the lemniscate r 2 = a 2 cos 2 0, 



ds a 4 
d6~ r' 



ip=2d+\7Z, — = -, and P=~ y 



2. Find the value of/ in the case of the curve r n = a n sin nd. 

p = a (sin nd) I+ ~n' 

3. In the case of the parabola referred to the focus 



2a 

r ■= 



1 _|_cos 6* 
prove that p 2 = ar. 

4. In the case of the equilateral hyperbola r 2 cos 26 = a 2 , prove 

that p = — . 
r 

d( 1 c^\ 

k. In the case of the ellipse r = — 7l , the pole beinsr at 

J L 1 — e cos * & 

the focus, determine/. a(i — e 2 ) 



\/(i — 2e cos -\- e 2 )' 

6. In the case of the cardioid r = a(i — cos 6), prove that 
tp = ^6, and that r 3 = 2ap 2 . 

a 2 

7. Trace the spiral r 2 =z — , which is known as the Lituus, finding 

the asymptote and the point of inflexion. 



312 APPLICATION TO PLANE CC/PF£S.[Ex. XXVII. 

8. Find the polar subtangent of the spiral r(e Q -\- e~ e )= a. 



e e - e~o- 
9. Find the subtangent and the subnormal of the spiral of Ar- 

ds A/(a 2 -{-r 2 ) 

chimedes, and prove that -- = — -. 

dr a 

10. Determine the asymptotes of the hyperbola from its polar 
equation referred to the focus, namely, 

a(e* - 1) 

r = — — 

1 — e cos 6 

X = ± sec~V, p — :f a \/(e 2 — 1). 

11. Prove that the condition which determines points of inflexion 

d^u 
in polar coordinates, namely, that u -J- — shall change sign, is equiv- 

du 

(dr\ 2 <Pr 
alent to the condition that r 2 -f- 2 1 — 1 — r— -^ shall change sign. 

12. Show that the curve rd sin 6 = a has a point of inflexion at 

2<2 

which r = — . 
n 

13. Show that the curve r# w = # has points of inflexion deter- 
mined by 6 = s/\jn{\ — w)]. 

14. Show that, if the curve r = "Q-J: has an asymptote whose 
inclination to the initial line is 6 V the perpendicular on it will be 

Ml 

except when^flj) is infinite. 

15. Show that the curve r(2 — 1) = 2<z6 has an asymptote de- 
termined by = \ , p = — \a, and a point of inflexion determined by 
the real root of the equation 2 # 3 — 6 2 — 2=0. 



§ XXVI I . ] EX A MPLES. 3 1 3 

1 6. The Conchoid of Nicomedes is defined as the protraction of a 
straight line, so that its polar equation is 

r = a sec ± b. 

(Compare Ex. XXIV. 20.) Show from this equation that the maxi- 

a 
mum ordinate corresponds to cos 3 ^ = — , and the points of inflexion 

to a root of 

b 2 cos 4 # -J- a & cos3 ^ — 4# 2 cos 2 # — tab cos 6 -f- 4a 2 = o. 

17. Trace the curve r = 0,(2 sin — 3 sin 3 /9). 

18. Trace the curve (x 2 -\-y 2 y — ^a 2 x 2 y 2 = o, first converting to 
polar coordinates. 

19. Trace the curve r 3 = a 3 sin 36^. 

20. Trace the curve r=a cos 3 i#, finding maximum ordinates 
and abscissae. 

21. Trace the curve r = 2 -|- sin \6, finding three double points. 
2 2 . Trace the curve r cos 6 = a cos 2 #, finding an asymptote. 

23. Trace the curve r cos 26 = a. 

24. Trace the curve r 2 = a 2 cos 26. 

25. Trace the curve r = a(cos # -f- cos 2 0), showing that there 
is a cusp at the origin and two double points. 

26. Trace the curve r 2 sin 6 = a 2 cos 2 0. 

27. Trace the curve r 2 cos = a 2 sin 3$. 

28. Trace the curve r= -^ : — ^ 

6> + sin 

29. The locus of the points the product of whose distances from 
two fixed points is constant is known as the Cassinian Oval. Show that 
the polar equation of the Cassinian, when the fixed points are (±a, o) 
and the constant product c 2 , is 

r 4 — 2a 2 r 2 cos 2 -f- a A — c 4 = o. 

Show that the curve becomes a lemniscate when c = a, and consists 
of a single oval surrounding the lemniscate or two ovals within its 



3 1 4 APPLICATION TO PLANE CURVES. [Ex. XXVII. 

loops, according as c>a or c<a. In the latter case, determine the 
values of 6 and r when r is tangent to the curve. 

sin 26 = ± -J ; r— ^{c& — c 4 ). 

30. Show that the rectangular equation of the Cassinian is 

thence show that the curve has maximum ordinates at points where 
r = a, and that double tangents and points of inflexion exist when 

2<Z>C>tf. 



XXVIII. 
The Measure of Curvature. 

325. Regarding a line as the path of a moving point, its 
curvature is measured by the rate at which the direction of the 
motion changes as the point moves uniformly along its path. 
Thus the curvature of a straight line is zero, because there is 
no change of direction. The curvature of a circle is the same 
for all its points, because if a point P moves uniformly in the 
circumference its direction (being always perpendicular to that 
of the uniformly rotating radius OP) changes at a uniform rate. 

For any curve other than the circle, the curvature varies 
from point to point. Denoting, as usual, the arc described by 
the moving point by ds, and by <p the inclination of the curve 

at P to the axis of x, — is the rate of change of direction, and 

dt 

ds 

t- is the rate of the point's motion. Hence the relative rate 

dt 

~ is the measure of the varying curvature. 
as 



XXVIII.] RADIUS AND CIRCLE OF CURVATURE. 3 1 5 



The Radius and Circle of Curvature. 

326. When the point P moves with a constant linear rate 
in the circle whose centre is O and whose radius is a, the arc 
described in the time dt is ds = adcp, because OP revolves with 

d<p . , 

the angular rate —r-, so that ds subtends the differential angle 
at 

d(p at the centre O. Therefore 



d(p 
d7 



that is to say, the measure of the curvature of a circle is the 
reciprocal of its radius. 

When the curvature is variable, the reciprocal of the meas- 
ure of curvature at any point is called the radius of curvature 
for that point, and is denoted by p. Thus 



9 



ds 
d4> 



(0 



is the radius of a circle of which the curvature is the same as 
that of the given curve at the given point. In Fig. 63, let 
PC, equal to the value of p for the 
point P of the curve AP, be laid off 
upon the normal on the concave side 
of the curve. Then the circle with 
centre at C, and radius equal to p, 
not only touches the given curve at 
P, but has the same curvature. This 
circle, represented by the dotted line 
in Fig. 63, is called the circle of curva- 
ture for the point P. If 5 is meas- IG * 3 * 
ured from the fixed point A, so that the direction AP is posi- 




3l6 APPLICATION TO PLANE CURVES. [Art. 326. 

tive, the figure represents the case in which <p increases 
with s, and therefore p is positive. Hence we infer that, when 
p is positive, it should be laid off from P in the direction 
-{- 90 . The point C so found is called the centre of curvature. 

327. In Fig. 19, p. 97, a group of curves is represented 
passing through a common point in a common direction. Of the 
two curves AB and A'B' lying above the tangent line, A'B' ', 
which has the least curvature, (and therefore the greatest 
radius of curvature,) lies between the curve AB and the tan- 
gent, on each side of the point of tangency. This will neces- 
sarily be the case, at least in the immediate neighborhood of 
the point of contact, when the radii of curvature differ in value. 
Thus, if a number of circles were drawn touching the given 
curve at P, in Fig. 63, any one with radius greater than p 
would lie on the convex side of the curve in the immediate 
neighborhood of P, and any one with radius less than p would 
lie on the concave side. 

Now, except at points where the radius of curvature has 
either a maximum or a minimum value, it will be increas- 
ing or decreasing in value as the moving point passes through 
the given position. For instance, in Fig. 63 it is represented 
as increasing as we pass through P in the direction AP. 
Accordingly, the curve lies on the concave side of the circle 
on the side of P toward A, and on the convex side of the circle 
beyond P. Thus the circle of curvature generally crosses the 
curve, as well as touches it, at the point of contact. 

328. When the rectangular coordinates of a curve are 
given in terms of a third variable, the value of p is readily 
expressed in terms of this variable. 

For example, in the case of the cycloid, equations (1), Art. 
278, the values of and of ds are given in equations (1) and 
(2), Art. 279. Substituting in the expression for p, 



§ XXVIII.] CURVATURE AT SPECIAL POINTS. 317 

ds 

P = a\p = ~4 asm *#• 

This result shows that p is in value double of the chord PR 
in Fig. 43. 

In particular, putting ip = o, we see that p vanishes at the 
cusp, and putting ?p = n, p has for the vertex O f its maximum 
value 4a. In accordance with the preceding article, the 
circle of curvature for the vertex will lie on the upper or 
concave side of the curve on each side of the vertex. A 
slightly smaller circle touching the curve at the vertex would 
lie below the curve in the immediate neighborhood of the 
vertex and would cut it in points on each side. 

The Radius of Curvature where the Curve is 
Parallel to one of the Axes. 

329. When = o, that is, at points where the tangent to 
the curve is parallel to the axis of x, we have dy = o and 
therefore ds = dx, and also d tan = d<fi. It follows that at 
these points 

d(p d tan d 2 y 
ds ~~~ dx ~~ dx 2 ' 

Denoting the radius of curvature at this particular point by 
p Q , we have then 

p ° = i (0 

dx 2 

In like manner, at a point where the curve is parallel to 

d 2 x 
the axis of y. the measure of curvature is -7—. Hence for a 

dy 2 



3i8 



APPLICATION TO PLANE CURVES. [Art. 329. 



" vertical " point of the curve, we have 

1 



Pi == 



d 2 x 



(2) 



The values of the special derivatives involved in these 
formulae are very readily found in the case of any algebraic 
curve. For we have seen, in Art. 168, that when (a, b) is a 
point of the given curve, 



y — b' 



x — a. 



dy' 
dx. 



— a, b 



But, if = o at the point {a, b) f the value of this expres- 

y — b . o 

sion is zero, so that -, - 2 takes the indeterminate form — . 

(x — ay O 

Evaluating this last expression, we have 



y -- b 
{x -- a)*J ati 



dy 
dx 



— b H dx 

- a) 2 \„ h 2(x - 



O I d 2 y~] 






Therefore ~r~ 2 = 2 ' 

Act,b 



dx' 



tion (1), 



{x— a) 2 J ati 
(x — a) 2 



a )_a,b O ' 2 dx 2 A a> ' b 

, and substituting in equa- 



Po 



2{y-b)J a>5 



(3) 



In like manner, we obtain for the vertical points of the 
curve 

{y-by 



Pl- 



2{x — a\\ a J 



■ (4) 



§ XXVIII.] LOCUS OF CENTRE OF CURVATURE. 319 

330. For example, in the case of the ellipse, 

x 2 y 2 

the curve is parallel to the axis of x at the extremity (o, b) 
of the minor axis; whence, for this point, by equation (3), 



x 

Po = 



2 -^ 



2(J> - *)Jo,*' 

Putting the equation in the form 



we derive 



a 2 

a\y -\- b)~ 



2(y - b)J 



o,3 2b" 



o, b 



a' 

T' 



In like manner, the radius of curvature at the point {a, o) is 



b 2 
found to be — . 
a 



The Locus of the Centre of Curvature, or Evolute. 

331. A clear conception of the mode in which the curvature 
of a given curve varies is best obtained by a consideration of the 
locus of the centre of the circle of curvature. Suppose P in Fig. 
63 to move along the curve, carrying with it a plane in which 
are drawn two fixed straight lines intersecting at right angles, 
and let these lines coincide at every instant with the tangent 
and normal at P to the given curve. Then the motion of this 
plane, as it slides over the fixed plane, consists of a rotation 
about P combined with the motion due to the linear motion of 



320 



APPLICATION TO PLANE CURVES. [Art. 33 1. 



P. Now the motion of this plane might equally well be de- 
fined as a rotation about any other point of it, combined 
with the linear motion of that point. Let us take, for this 
purpose, that point of the moving plane which, at the in- 
stant represented in the figure, is situated at C. This point 
has no motion at the instant; for, if each of the two com- 
ponent motions first mentioned became uniform, the motion of 
the plane would become simple rotation about C, P then de- 
scribing the circle of curvature. C is therefore called the 
instantaneous centre of the motion of the plane. 

It follows that the motion of the centre of curvature in the 
fixed plane is simply that due to the change in the value of p ; in 
other words, C moves in the direction of the normal to the given 

curve, and at the rate — , which is its rate in the moving plane, 

(mi/ 

as it moves along the line PC of that plane. The curve de- 
scribed by C is called the evolnte of the given curve, and the 
motion of the plane may now be defined by the trolling of the 
line PC drawn in it upon the evolute drawn in the fixed plane. 
332. As an illustration, the evolute of the ellipse is drawn 




in Fig. 64. The values of AC, 

and BC 2 > the radii of curvature at 

b 2 
the extremities of the axes, are — 

a 



and -=- respectively, as found in 



Art. 330. As P moves from A to 
B in the ellipse, C moves from C x 
Fig. 64. to C 2 , describing an arc touching 

each axis. Similar arcs correspond to the other quadrants of 
the ellipse. 

It will be noticed that cusps of the evolute correspond to 



§ XXVIII.] RADIUS OF CURVATURE. 321 

maxima and minima values of p. If a solid piece having the 
arc Cfi 2 for its convex outline be made, we can suppose the 
quadrant BA of the ellipse to be described by the extremity 
of a string C 2 B which is wound upon the arc C 2 C .* 

The Radius of Curvature in Rectangular Coordinates. 

333. To express p in terms of derivatives with reference 
to x, we have 



£ = i / [ I +(£) 2 J: and ^ = tan 



whence 



d 2 y 
d(p dx 2 



dx ~ , (dy \ 



2 » 



1 v dx J 



therefore 



ds 
dx 



'+( 



dy^- 
dx) 



? 



dcf) d l y 

dx dx 2 



(1) 



(Py 
In this expression, p has the sign of -~ , and therefore, by 

, il/X^ 

Art. 98, has the positive sign when the curve is concave as 
viewed from above, and vice versa. This will be found to 
agree with the rule for the direction of p given in Art. 326. 

* In the approximate construction of an arc of a curve by means of circular 
arcs, in mechanical drawing, the evolute of the arc to be drawn is practically 
assumed to be a polygon instead of a curve. 



322 APPLICATION TO PLANE CURVES. [Art. 334. 

334. For example, to express the radius of curvature of 
the ellipse 

a 2 ^ b 2 
in terms of the abscissa, we have 





y = ± -*J(a 2 — x 2 ). 


Differentiating, 






dy bx 




dx " a \/{a z — x 2 )* 


and 






d?y ab 



**■ (a 2 -x 2 f 

Thus 1 + l£) = ^ _ x2) , and substituting in equa- 
tion (1), Art. 333, 

[a 4 - (a 2 - b 2 )*?Y 
^ =qF aW ' 

the upper sign corresponding to the upper or convex semi- 
ellipse, and the lower to the concave half. Supposing b < a> 
and putting |/(a 2 — b 2 ) = ae, where e is the eccentricity of the 
ellipse, this becomes 

( fl » - eVf 



§ XXV ill.] RADIUS OF CURVATURE, 323 

335. A more symmetrical expression for p is obtained by- 
using derivatives with respect to a third variable, say t; we 
then have (compare equation (2), Art. 86) 

dy dx d 2 y dy d?x 



, _\dt 
<fi = tan 

dx ' 

dt 


deb dt dP dt dt 2 


Therefore 






mmry 




dx dy dy d 2 x 



dt dt 2 dt dt 2 

If in this expression / denotes time, it gives the value of 
p in terms of the component velocities and accelerations of P 
along the axes. By Art. 326, this value when positive is to be 
laid off on the left hand of the direction in which the point is 
moving. 

When y is the independent variable, we may write 

. (dx\ If 
L 1 + \d-y) J 

p = ^T ' (3) 



dy 2 

which is positive when the concave side of the curve is on the 
right, just as expression (1), Art. 333, is positive when the con- 
cave side is upward. 

336. When s is taken as the independent variable, the 
numerator of expression (2) becomes unity, and the denomi- 
nator, which is now the value of the measure of curvature, is 

1 _ dx d 2 y dy d?x 

p ds ds* ds ds 2 ' x * 



324 APPLICATION TO PLANE CURVES. [Art. 336. 

But this expression may be simplified by means of the relation 

©'+(£) = « 

and its derivative 

dx d 2 x dy d 2 y 

d^~d? + did? = (3) 

d/x d?y 

Eliminating successively -r-j and -~ from equation (1), we de- 



rive 



dx dy 

ds ds . 

P =d*y=--<Pi • (4) 

ds 2 ds* 



Again, eliminating the two first derivatives, we obtain 

J z ~\ds i ) ~T~\ds 2 ) ' 



The Equation of the E volute. 

337. Denoting the coordinates of C in Fig. 63 by x r and 
y, we have, by projecting upon the axes the line PC = p, 
whose inclination to the axis of x is -j- 90, 

1 dy 
xt = x — psm = x — p-j-y . . . (1) 

dx 

? = y + p cos <p = y + p-fc- • • • (2) 



§ XXVIII.] THE EQUATION OF THE EVOLUTE. 325 

Hence, using equation (1), Art. 333, 

X = x — -f> * ~r~ * • • • • (3/ 

dry dx 

dx 1 

2 



■ + m 



y=y+ — d^— (4) 

dx 2 

If, in these equations, y and its derivatives are expressed 
in terms of x by means of the given equation, we have only to 
eliminate x to obtain the equation of the evolute. 

For example, to find the evolute of the common parabola 
y 1 = 4ax, we have 



h h , dy or 

y = 2a x , whence -^ = — =■ 

dx ~$' 



i+ (2) =0 4^ and &* 

Substituting in equations (3) and (4) , 






(fry a 



2X* 



2x^ 

x' = 2a+3x, y f = I - t 

a? 



and eliminating x, we have 



27a/ 2 = 4(xf — 2af. 



326 APPLICATION TO PLANE CURVES. [Art. 337. 

Therefore the evolute of the parabola is a semi-cubical parab- 
ola (Art. 250), having its cusp at the point (2a, o). 

338. One or both of the equations (3) and (4) may be 
replaced by similar equations in which y is the independent 
variable, so that x and y are interchanged throughout. For 
example, using the results obtained in Art. 334, equation (3) 
gives for the ellipse, 

x 2 v 



tf x - {a 2 - b 2 )x 3 (a 2 - b 2 )x 3 
x' — x = 1 ... (2) 

Since the equation of the ellipse is unchanged when we inter- 
change x and y and at the same time a and b, we infer at once 
that the corresponding equation with y as independent variable 
would give 

/ = — ~ t — • ....... (3) 

Eliminating both x and y between equations (1), (2) and (3), 
we obtain 

(**')*+ (by'f = (a 2 - b 2 f y 

which is therefore the equation of the evolute drawn in Fig. 64. 

ds 
339. Substituting -=- for p in equations (1) and (2), Art. 

337, we obtain the formulae 

dy dx 



XXVII.] THE E VOLUTE OF THE CYCLOID. 



327 



which are convenient when x and y are expressed in terms of 
a third variable. For example, in the case of the cycloid, 



x = a(tp — sin ip), 
we have, as in Art. 279, 

dx = a{\ — cos tf>)dip, 



y = a{\ — cos i/>), 



dy = a sin ipcfy 



and = 90 — J#, whence d<fi = — \dty. Substituting in the 
formulae, we find 

x' = a{$> + sin if>), y' — — a {\ — cos ^). 

Comparing with the equations of the cycloid referred to its 
vertex, Art. 280, we see that the evolute is a similar cycloid 
below the axis of x, as in Fig. 65 . 

340. This result may also be 
derived geometrically from the 
fact, proved in Art. 328, that the 
radius of curvature is double of 
the chord PR of the generating 
circle. For it follows that Q, the 
centre of curvature, is a point of 
the equal circle RQR' situated 
below the axis of x. Hence the arc R'Q is the supplement 
of PR and therefore equal to the distance 0' R' , so that the 
circle RQR' rolling upon the lower horizontal line generates 
the locus of Q, which is thus a cycloid equal to the given 'one. 

If solid pieces having the arcs of the lower cycloid as 
their convex outlines be employed, as in Art. 332, the upper 
cycloid may be described by the extremity of a string of 
length r A, having the extremity O' fixed, and being wrapped 
upon the arc O'O and O'B. It follows that the arc 00' is 




Fig. 65, 



328 APPLICATION TO PLANE CURVES. [Art. 340. 

equal to OA = 4a; so that the whole length of a branch of the 
cycloid is eight times the radius of'the generating circle. 

If the figure be inverted it represents the method in which 
Huyghens proposed to make a particle oscillate in a cycloid, 
to illustrate his discovery of the isochronism of these vibrations. 



The Radius of Curvature in Polar Coordinates. 

34-1. When the curve is given by means of its polar equa- 
tion we have, Fig. 51, Art. 297, = ip -\- 0, whence 



ds ds 

P = 

where 



p - d<p- d6 + dp (I ) 



ds 2 = dr 2 + r 2 d&, (2) 

and 

rdO 
tan1p = ~aV (3) 

In differentiating equation (3) to obtain an expression for dtp, 
dS may be regarded as constant ; since the result is to be ex- 
pressed in derivatives with reference to 0. Hence 



dr 2 — rd 2 r 
~dr< 



sec 2 tpdtp = ^2 dO, 



ds 
and, since sec ip = -j- s 



dr 2 — rd 2 r . 
d*/> = - -772 dO. 



ds 
Hence 



dr 2 + ds 2 — rd 2 r ,„ 

d& + dip = — ^—^ de, 



XXVIIL] 



POLAR COORDINATES. 



329 



and, substituting in equation (1), we obtain 

ds 3 



therefore 



or 



P 



P = 



(dr 2 -j-ds 2 - rd 2 r)dd y 

(dr 2 4- r 2 dd 2 f 
(2dr 2 + r 2 dd 2 — rd 2 r)dd s 

r/dr^ 2 



P = 



dO 



+ r< 



r^A^~r% 



dd 



dd 2 



(4) 



342. To obtain p in terms of u, Art. 298, we eliminate r 
from equation (4) thus: 



and 



1 </r 1 afo 

r=-, then -75= r-^, 

« tf # u 2 dd 



d 2 r 2 /d# \ 2 1 d 2 u 



On substituting these values, we obtain 



P = 



2 v 3 



»-+(sr* 



zrl & -\- 



df 2 . 



(5) 



It will be noticed that the denominator of this value of p 
contains the expression (1), Art. 318, which changes sign, 
and is therefore usually equal to zero, at a point of inflexion. 




33° APPLICATION TO PLANE CURVES. [Art. 343. 

Relations between p, p and r. 
343. In Fig. 66, if we denote OR by p and PR by r, we 



p = r sin i/) t and r = r cos ip. (1) 

Now let P move along the curve at the rate 

ds 
a — , then the tangent PR will rotate about P 

D ^ d(p 

Fig. 66. at t ^ ie an gul ar ra te -77, and OR will rotate 

about O at the same rate, because these lines are always at right 

angles to each other. The motion of the point R* may be 

resolved into two motions: one in the direction OR, and the 

other in the direction RP. Since the velocity of P in the 

direction OR is zero, the component of the velocity of R in this 

d(/> 
direction is r — — , due to the rotation of PR about P, while 
at 

d<P 
the component in the direction RP is p-rr- The first of these 

components is the rate of p, since O is a fixed point ; therefore 
dp d<P dp 

_ =r _ whence r= _ . . . (2) 

The rate of r is the difference between the velocity of P in 
the direction RP, and the component velocity of R in the 
same direction ; therefore 

dr ds d<P 7,7 

dF = dT- p ti' ° r dr = ds-pd4,. . (3) 

* The locus of R is called the pedal (Art. 313) of the given curve from the 
origin O. The relation between p and <p gives the polar equation of the pedal, 
<p being the angular coordinate of the radius vector p when measured from an 
initial direction 90 behind OA, that is, downward in Fig. 66. 



§ XXVII I. ] INTRINSIC EQ UA TIONS. 331 

344, By comparing the expressions for r in equations (i) 

dr 
and (2), and putting for cos tp its value—, we obtain 

dp dr 

dcf> ds ' 
whence 

ds rdr 

p = T$ = dp • (4 > 

An expression for p may also be derived from equation (3) ; 
thus, 

ds dr 



or, by equation (2), 



d 2 p 
p = ? + d^ P> 



Intrinsic Equations. 

345. The relation between 5 and for any curve is called 
its intrinsic equation, because it is independent of any geomet- 
rical elements exterior to the curve, except the direction (p— o, 
from which is measured, and the point on the curve from 
which s is measured. The intrinsic equation is usually put in 
the form s == f(<j>) ; and, by preference, is so taken that s van- 
ishes with <j), so that the initial direction (or <f> = o) is that of 
the tangent to the curve at the initial point. Thus the intrin- 
sic equation of the circle is written s = acp, and the addition of 
a constant only changes the point from which 5 is measured. 

Differentiation of the intrinsic equation gives at once the 
value of p in terms of ; on the other hand, the determination 
of s in terms of from the value of p, or of ds (as derived from 



332 APPLICATION TO PLANE CURVES. [Art. 345, 



the rectangular equation of the curve), requires in general the 
inverse process of Integration. Curves for which simple ex- 
pressions for 5 exist are said to be rectifiable y and it is these 
curves which have simple intrinsic equations. 

346. For example, in Fig. 65, we have seen that the arc 
OQ of the lower cycloid is equal to PQ or 2RQ. Reckoning 
from the initial direction OB toward the right, is the 
angle QRB, and RQ = 2a sin <p ; therefore 

s = 4a sin (1) 

is the intrinsic equation of the cycloid referred to its vertex. 
This equation gives a maximum value for s, namely 4a, when 
— \n at the point '. Differentiation of equation (1) gives 

ds = 4a cos0 dcp (2) 

Hence, if continues to increase beyond the value \n, ds 
changes sign, so that, while Q describes the arc O'B, s in equa- 
tion (1) is the algebraic value of the arc with the new branch » 
reckoned as negative. Thus s is again zero at the vertex B y 
and becomes — 4a at the next cusp. 

347^ As another illustration, let us find the intrinsic 
equation of the catenary (see note p. 217) of which the rect- 
angular equation is 

y , = \([ c + ■'" ~ 7 ) (I) 

Here 

_Z — _ ( e c e c \ — tan 0, . . (2) 

dx 2 



XXVIII.] RADIUS OF CURVA TURE OF THE E VOLUTE. 333 



and 



f 1 / - --V1 dx*/ - --\2 



Hence 



ds 1 / - , ~-\ , v 

^=ir+* v (3 > 

and it is clear that, if we measure s from (o, c), the point A in 
Fig. 67, p. 334, we must have 

X x 

s — -ie c — e c J, (4) 

because this gives equation (3) by differentiation and makes 
s = o when ^r = O. Comparing with equation (2), we have 

s = c tan (5) 

for the intrinsic equation of the catenary. 



The Intrinsic Equation and Radius of Curvature 

oj the E volute. 

348. If s' and <fi' denote the intrinsic coordinates of the 
evolute, we have seen in Arts. 332 and 326 that 

ds' = dp, and <ft r = -|- 90 , whence d<fi' = d<p. 

It follows that s' and p can differ only by a constant depend- 
ing upon the initial point from which the arc is reckoned, so 
that the value of p in terms of gives the intrinsic equation 
of the evolute. 



334 



APPLICATION TO PLANE CURVES. [Art. 348. 



For example, the intrinsic equation of the catenary, equa- 
tion (5) above, gives p = c sec 2 0. 
The minimum value c occurs at 
the vertex A, Fig. 67, giving the 
cusp A' of the evolute. If then 
we measure s' from the cusp, we 
shall have s' — p — c = c tan 2 0; 
and measuring 0' from the verti- 
cal direction, so that <fi = 0', we 
have 




Fig. 67. 



s f = c tan 2 0' 



(6) 



for the intrinsic equation of the evolute. 

349. Since p\ the radius of curvature of the evolute, is 
equal to ds' '/dtp* ', we have 



ds 



, __df> _ d?s 



d(p dcfj* * 
For example, for the evolute above, we find 

p' = 2c tan sec 2 0. 

In like manner, given the intrinsic equation of any curve, 
we can find the intrinsic equations and the radii of curvature of 
its successive evolutes. 



Involutes and Parallel Curves* 

350. When a tangent line rolls upon a given curve, any 
point on it describes a curve of which the given curve is the 
evolute. Such a curve is called an involute of the given curve, 
and other points on the tangent describe other involutes of the 



§ XXVIIL] INVOLUTES AND PARALLEL CURVES. 335 

same curve. For example, the lower cycloid in Fig. 65 being 
given, the upper cycloid is the involute described by the point 
P, and the dotted line is the involute described by another 
point of the rolling tangent. The involutes of a given curve 
form a system of curves having common normals. With ref- 
erence to one another, they are called parallel curves because 
the corresponding points of any two of them are at a constant 
distance measured along the normal. 

351. Denoting this constant difference by c, if s = f(<t>) is 
the intrinsic equation of a given curve, that of the parallel 
curve is s = /"(0) -|- c0. For the radius of curvature of this 
curve is 

and that of the given curve is f((f>). If (x, y) and (x v y x ) are 
corresponding points on the two curves referred to rectangular 
axes, the inclination of c to the axis of x is -\- 90 (in 
Fig. 65, taking the cycloid as the given curve, c is negative); 
hence, projecting it upon the axes, we have 

dy 
x, — x = — c sin d> = — c~tj 
1 as 

dx 
Ji — 7= ccos 0= c-^. 

Hence the equation of a parallel to a given curve is the result 
of eliminating x and y between these two equations and that 
of the given curve. 

35 2u From the construction of the involute of a given 
curve it is obvious that a cusp occurs whenever the involute 
meets the curve at an ordinary point (that is at a point which 
is neither a cusp nor a point of inflexion). In Fig. 67, that 



336 APPLICATION TO PLANE CURVES. [Art. 352. 

involute which meets the catenary at the vertex A is drawn ; so 
that, if p x denotes PP lt the radius of curvature of the involute, 
we have p } = s. Equations (1) and (3), Art. 347, show that 
y = c sec 0. Hence if we join R, the foot of the ordinate, 
with P lt and compare equation (5), we see that PP^R is a 
right angle. Therefore RP k is tangent to the involute, and is 
equal to the constant c. It follows that this curve is the path 
of a heavy particle on a horizontal table when attached to the 
end of a string of length c, the other end of which is moved 
along the line OX. The curve is, for this reason, called the 
Tractrix. 

353. A curve can have involutes which fail to meet it only 
in case there is a portion of the rolling tangent which the 
point of contact never reaches. For example, in Fig. 67, the 
point of contact P' moves along the rolling tangent PCP f y but 
never passes beyond the fixed point C, at a distance c from 
P. Any point on the same side of C with P describes a curve 
without cusps. Points on the other side obviously describe 
curves with two cusps. 

Again, in Fig. 65, the point of contact travels back and 
forth over a segment of the rolling line equal in length to one 
branch of the cycloid, and points beyond this segment de- 
scribe looped curves without cusps. So also, in Fig. 64, the 
point of contact is confined to a segment equal in length to one 
branch of the cusped curve of which the ellipse is an involute. 
It is only in such a case as this, when the algebraic sum of the 
arcs of the given cusped curve is zero, that the involute can be 
a closed oval. 

354. On the other hand, the involute of a closed oval 
will be a spiral having one cusp, and only one. For example, 
Fig. 68 represents the involute of the circle whose radius is a. 
In this particular case, all the involutes are evidently alike in 



XXVIII.] INVOLUTES AND PARALLEL CURVES. 



337 



shape; in other words, the involute of the circle is identical 
with its own parallels, a prop- 
erty shared only by the straight 
line. 

In Fig. 68, let O be taken as the 
origin of rectangular coordinates, 
and OA as the axis of x ; then de- 
noting A OB by i/>, BP = aip, and 
projecting OBP upon the axes, we 
have 




Fig. 68. 



x = a cos ip -f~ afy sm ^» ) 
y = a sin tp — aip cos tp, j 

for the rectangular equations of the involute of the circle, 
The intrinsic equation of this curve is 

s — iacp 2 , 

where for the point P is the same as ip in Fig. 68; for this 
equation gives for the radius of curvature p = a<p. 



Examples XXVIII. 

i. Find the radius of curvature of the parabola 

Vx , Vy 

= i 

Va Vb 

at the point where it touches the axis of x. 



2<2 4 

A,= — 



2. Find the radius of curvature of the four-cusped hypocycloid 
x =■ a cos 3 ip, y = b sin 3 ^. 

p == — 2> a s ^ n *p cos i>' 



33 8 APPLICATION TO PLANE CURVES. [Ex. XXVIII. 

3. Find the radius of curvature of the three-cusped hypocycloid 

x = 0(2 cos ip -\- cos 2ip), y = a{2 sin — sin 2ip). 

p = — 8a sin -|^\ 

4. Find the radius of curvature of the curve 

x = 2a sin 2ip cos 2 ip, y = 2a cos 2ip sin 2 ^?. 

p = 4a cos 3^. 

5. Find the radius of curvature of the parabola y 2 = ^ax. 

(<2 -f- Jf)^ 



p= if 2 



6. Find the radius of curvature of the catenary 



y = C -{e« + e A, 

and show that its numerical value equals that of the normal at the 
same point. y 2 



7. Find the radius of curvature ot the semi-cubical parabola 

ay 2 = x 3 . 

(4a -f- gx) x 



1 I 



p = 



6a 
8, Find the radius of curvature of the cissoid 



3 

■g- 

X 


p = 


a \/x{8a - 
3 ( 2 a - 




{2a — x) 


-3*)' 

x) 2 * 



§ XXVIII.] examples. 339 

9. Find the radius of curvature of the parabola 

\/x -f- |/j> = 2 |/<Z. 

3 

{x+y? 

\>'a 
10. Find the radius of curvature of the logarithmic curve 



y = ae c 



1 j — 



cy 

11. Find the radius of curvature of the hyperbola 

a 2 T =? ' 

(e 2 x 2 — «2) 

12. Find the radius of curvature of the cubical parabola 



a 2 y = ^r 3 . 



(^+9^) 



6<2 4 X 

13. Find the radius of curvature of the prolate cycloid 
x — arp — b sm tf: , y = a — b cos tp. 



__ (a 2 - \- b 2 — 2ab cos ip) 
b(a cos t/j — 3) 

14. Find the radius of curvature of the rectangular hyperbola 



xy = m 2 . 



_ (x 2 +f) 
fj — 



2171" 



340 APPLICATION TO PLANE CURVES. [Ex. XXVIII. 

15. Show that when n > 2, the radius of curvature of the parab- 
ola of the ttth degree is infinite at the origin, and is a minimum 
where 

y_ V( n ~ 2 ) 
x nt^{n — 1) ' 

16. Given the curve y % -f- x* -\- a(x 2 -\- y 2 ) =a 2 y. Find the value 

of p at the origin. _ a 

Po - -• 

17. Given the curve ax 3 — 2b 2 xy -j- cy 3 = x i -j- y*. Determine 

the values of p at the origin. 

b 2 
For the branch tangent to the axis of x, p Q = — ; 

for the branch tangent to the axis of y, p x = — . 

c 

18. In the case of the strophoid, Fig. 40, p. 257, find the value 
of p at the vertex ; also, after turning the axes through 45 °, the value 
of p at the origin. \a ; £ 4/2 . a. 

19. Given the curve x* — ax 2 y ~\- axy 2 -f- \a 2 y 2 = o. Determine 
the value of p G . See Ex. XXV. 19. p Q = \a. 

20. Find the radius of curvature at the origin, the equation of the 
curve being 

x 4 — \ax 2 y — axy 2 -\- a 2 y 2 = o. 

p Q = a and p = \a. 

21. Given the curve x 5 — ^ay* -f- 2ax 3 y -|- a 2 xy 2 = o. Find the 
values of p . See Ex. XXV, 20. 

p Q = — \a, at the cusp; 

p = I0, for the other branch. 

22. Find the equation of the evolute of the hyperbola 

x 2 y 2 
a Y ~~¥ =I ' 

(axf — {by? = (a 2 + b 2 ?. 



§ XXVIII.] EXAMPLES. 341 

23. Prove that, in the case of the four-cusped hypocycloid 
x = a cos z tp, y = b sin 3 ^>, 
the coordinates of the evolute satisfy 
x' -\-y r = a(cos t/j -J- sin ip) 3 and x —y = a(cos tp — sin ip) z , 



and thence deduce the rectangular equation of the evolute. 
24. Given the equation of the catenary 



(■*' +yf + {*' - y'f = 2/ 



y = a -(e a •: 



prove that 

y = 2y, and x' = x — —le a -\- e a ), 

and deduce the equation of the evolute. (See Fig. 67.) 

± x> = a log y+Q'"-^)* _ y ^ _ j 

2a \ar ' 

25. Find the equations of the evolute of the curve 

x = c sin 2//;(i -J- cos 2tp), y = c cos 2^(1 — cos 2^). 

x' = 2c(— 2 sin tp cos 3 ^-j- sin 2tp cos 2 ip), 

y — 2C(2 COS tp COS 3//J -f COS 2>p sili 2 tp). 

26. Find the radius of curvature of the limacon 

r = a -\- b cos 6. 



(a 8 + 2g£cosfl + ff )' 

^ 2 + 3^ cos # -J- 2b 2 ' 

a 2 
27. Show that, for the lemniscate r 2 = a 2 cos 2 0, p = — ; and 



r 3 

that, for the equilateral hyperbola r 2 cos 2 # = a 2 , p = . 

a 2 



34 2 APPLICATION TO PLANE CURVES. [Ex. XXVIII. 

28. Find the radius of curvature of the conic referred to its focus 

a(i — e 2 ) 






— e cos 6' 

_ a(i — e 2 )(i — 2e "os 6 -f- <? 2 ) 



(i — £ cos #) 3 

29. Find the radius of curvature of the lituus 

r 2 6 = a 2 . 

p= r( > 4 4- r 4 ) 1 
2^ 2 (4« 4 — r 4 )' 

30. Given r w = a m cos #z#, prove that r m + 1 = a m p, and thence by 
means of equation (4), Art. 344, prove that 

r 2 
P 



{m-\- i)p 

31. From the relation between p and r in the case of the parabola 
referred to its focus, Ex. XXVII. 3, prove by means of equation (4) 
Art. 344 the following construction for the radius of curvature of the 
parabola: Join P to the focus F and draw FN perpendicular to FP 
to meet the normal at P in N, then p = 2PN. 

32. Show that the rectangular equations of the catenary in terms 
of a third variable are 

x = c log (sec -f- tan 0), y = c sec 0. 

33. Show that the intrinsic equation of the cycloid as measured 
from a cusp is 

s == 4<z(i — cos 0) = 8a sin 2 10. 

34. Find the radius of curvature of the epicycloid 

d I <? CL I 3 

jp = (a -|- 3) cos tp — b cos — 7 — if>, y = (a -\- b) sin tp — b sin — - — tp, 

in which tp = corresponds to a cusp. 

A.b(a 4- $) . arp 

P — i — r~ sm "T- 

^ tf+20 20 



§ XXVI II. J examples. 343 

35. Using the general equations of the epi- and hypocycloids 

/? 7? 

x = R cos -I cos mcfi, y = R sin <b A sin md>. 

m m 

Art. 285, in which ip = o corresponds to a vertex, show that 

p = ■ — ■ cos \{m — i)0. 

36. Show that the intrinsic equation of the curve in Ex. 35 when 
= o is the direction of the tangent at the vertex, is of the form 

j = / sin 720, 

where « = = ;, so that n < 1 for the epicycloid, and 

m + 1 a + 23 r ^ 

« > 1 for the hypocycloid. Hence show that the evolute is a similar 

curve of n times the linear dimensions of the given curve. 

37. Denoting by r the number of cusps of an uncrossed epicycloid 
or hypocycloid, show that in the notation of Ex. 36 we have re- 
spectively 

n = and n = 



r + 2 r — 2 

38. Show that the intrinsic equation of the evolute of the four- 
cusped hypocycloid s = / sin 2 0, when measured from the cusp, is 
s' = 4/ sin 2 0, and transform this equation to its own vertex and 
tangent. 

39. Prove that the centre of curvature of the equiangular spiral, 

Art. 324, is the intersection of the normal and a perpendicular to r 

drawn through the pole; and thence that the evolute is a similar 

spiral which is identical with the given spiral turned forward through 

, n log n 

an angle . 

2 n 

40. Show that the intrinsic equation of the equiangular spiral is 

s = le n cp, 

where / is the whole length of the curve measured from the pole 
given by = — 00 to the point where = o. 



344 APPLICATION TO PLANE CURVES, [Ex. XXVIII. 

41. From the polar equation of the cardioid 

r= 2a sin 2 -J0, 
equation (1), Art. 312, derive the radius of curvature 

Ad . 

p == — sin W, 
3 

also the intrinsic equation referred to the cusp 

s = 40(1 — cos J0), 

and thence show that the evolute is a cardioid of one-third of the size 
rof the given one. 

42. Show that, in general: to an infinite branch corresponds a 
parabolic branch of the evolute ; to a point of inflexion an asymptotic 
branch ; and to a cusp a branch passing through the given point. 
What exceptional cases can occur? 



XXIX. 

Systems of Curves, 



355. The constants to which arbitrary values may be as- 
signed in the equation of a curve are called parameters of the 
curve, and the curves obtained by giving different values to 
the parameters are said to constitute a system of curves. 

When a single parameter is considered and that admits of 
an infinite number of values (that is to say, of continuous varia- 
tion), the system is called a singly infinite one. Denoting the 
variable parameter by a, the general equation of a singly 
infinite system 'is of the form 

/(#, y, a) =0, . . . . . . (1) 



§ XXIX.] SYSTEMS OF CURVES. 345 

in which different values of a distinguish different members of 
the system. If we suppose a to vary through all its possible 
values, the curve represented by equation ( ) sweeps over the 
whole or a portion of the plane. In fact, if we select any 
point P v and assume that its coordinates (x lt y x ) satisfy equa- 
tion (i), we have the equation f[x v y v a) — o, by which to 
determine the particular value or values of a for which the 
curve passes through the selected point. 

356. If the equation is of the first degree in a, so that it 
can be put in the form 

F i( x > J) + a F 2 (x, y) = o, . . . . (i) 

where F 1 and F 2 are one-valued * functions of % and y, we 
shall have, in general, for a selected position of P 1 a single 
value of a ; so that one, and only one, member of the system of 
curves passes through the selected point. But, if the curves 

Fi( x > y) = °i F 2 ( x > y) = ° 

intersect, each of the points of intersection will satisfy equation 
(i), independently of a, so that all the members of the system 
will pass through each of these points. It follows that the 
members of the system of curves intersect each other in no 
other points except these fixed ones, and that as a passes by 
continuous variation through all values from — oo to -|- oo the 
curve represented by equation (i) sweeps once over the whole 
plane. 

357. A system of curves of this kind is called a pencil of 
curves. For example, the circles with centre on the axis of x 

* Thus an equation containing radicals must be rationalized with respect to x 
and y, before its degree in a can be ascertained. 



34-6 APPLICATION TO PLANE CURVES. [Art. 357. 

and cutting the axis of y in the point (o, b) form such a sys- 
tem. For, if the centre of one of these circles is at (a, o) its 
equation is y 2 -f- (x — a) 2 = b 2 -f- a- 2 , or 

r^ + y 2 ~ 2fl,:x; — ^ 2 — ° ( x ) 

The system of circles resulting from making <* an arbitrary 
parameter all pass through the two points (o, b) and (o, — b) t 
and may be described as the pencil of circles passing through 
these points, or having a common chord on the axis of y. 
Accordingly, the equation is of the first degree in a, and the 
fixed points of intersection are the intersections of 

x 2 -\- y 2 — b 2 = o and x — o. 

If we change the sign of b 2 in equation (1) we still have a 
pencil of circles ; the axis of y is in this case no longer a com- 
mon chord, but in either case is called the radical axis of the 
pencil. 

Envelopes. 

358. In case the equation of the system is other than of 
the first degree in or, the curves of the system may intersect 
one another in other than fixed points. Let us suppose in the 
first place that 

f(x, y, a) = o (1) 

is of the second degree in a. Then, when the coordinates 
(x, y) of a selected point P are substituted in equation (1), we 
have a quadratic equation for a. If the roots of this equation 



§ XXIX.] 



ENVELOPES. 



547 




are real we find two values for a, determining two distinct 
curves of the system which intersect in P. Let P in Fig. 69 
be such a point, and PA, PB the two curves of the system 
thus determined. 

Suppose also that there are positions of P for which the 
values found for a are imaginary, so that no curves of the sys- 
tem pass through these points. 
Then, as P is moved from a posi- 
tion for which a is real to one for 
which a is imaginary, it will pass 
through a position for which the 
two values of a. are equal. The 
locus of the points which give 

equal values of a will, in fact, > 

constitute a boundary line sepa- 
rating a portion of the plane in 
which a is real from one in which 
it is imaginary. 

As P is moved up to the boundary line the two curves 
which there intersect come into coincidence. For this reason, 
the point on the boundary line is called the ultimate intersection 
of consecutive curves of the system. 

In general, the ultimate intersections take place at ordinary 
points of the curves of the system, that is to say, at points 
which are not double points or cusps. When this is the case, 
as in Fig. 69, the boundary line touches the curves of the 
system, and is called the envelope of the system. 

It follows that the envelope of a system of curves is the 
locus of the ultimate intersections or points for which a has equal 
roots. 



Fig. 69. 



34-8 APPLICATION TO PLANE CURVES. [Art. 359. 

Equations of the Second Degree in a. 

359. When the equation of the system of curves is of the 
second degree in a, it may be written in the form 

Pa 2 + Qa + R = o, (1) 

where P, Q and R are, in general, functions of x and y. 
The discriminant of this equation (or quantity which appears 
under the radical sign when it is solved for ot) is Q 2 — 4PR\ 
hence all points for which the values of a are equal must 
satisfy the equation 

Q 2 — 4PR = o. ...... (2) 

Consider now the locus of this equation in x and y. If in 
passing over any branch of this locus the discriminant changes 
sign, as will generally be the case, it will form a boundary 
between regions of the plane in which a is respectively real 
and imaginary, that is to say, it will be the locus of ultimate 
intersections and in general an envelope of the system. But 
if the discriminant does not change sign,- the branch will not 
form a boundary, and will fail to give an envelope. 

360. For example, the equation of the path of a projectile, 
neglecting the resistance of the air, is 

/y.2 

y = x tan a — ^7 =— , . . . . (1) 

J 4H cos 2 a ; 

representing a parabola in a vertical plane, the origin being 
the point of projection, and a the angle of projection. Sup- 
posing a to vary while H remains fixed, we have a system of 
parabolas in the same vertical plane, being the trajectories 
described with a given initial velocity. It is required to find 
the envelope of this system. Equation (1) is virtually one of 



§ XXIX.] EQUATIONS OF THE SECOND DEGREE IN a. 349 

the second degree in the arbitrary parameter; for, putting a 
for tan «, it may be written 

4JS(y — xa) + (1 + a 2 )x 2 — o, 
or 

x 2 a 2 — 4Hxa -f- 4H y -\- x 2 = o. 

Comparing with equation (1) of the preceding article P = x 2 , 
Q =z — 4HX and R = 4Hy -f- x 2 \ hence equation (2) gives 

. x 2 [4H 2 - UHy + x 2 )-] = o 

for the locus of points for which a has equal values. Here 
we reject the squared factor x 2 because, this factor being 
always positive, the discriminant does not change sign * when 
we cross its locus x = o. The other factor gives the envelope 

x 2 = 4H(H - y), 

* So in general we reject a factor of the discriminant which appears with an 
even exponent. In this example, the locus of the squared factor, x 1 = o, is a 
member of the system, corresponding in fact to a. = 00 (when the angle of pro- 
jection is 90 ). 

A squared factor will also occur in the discriminant when all the curves of the 
system possess a double point or node. For, in that case, the two branches which 
pass through a selected point P (belonging, in general, to different curves of the 
system) will, when P is brought up to the node-locus, become branches of the same 
curve, and so correspond to a single value of a. Thus the node-locus is part of 
the locus for which two values of a have become equal; but the values remain real 
on both sides of the locus. In like manner, an isolated point or acnode gives rise 
to a squared factor in the discriminant and to a locus on which the values of a 
are equal, but the values are imaginary on each side of the locus. 

When all the members of the system possess a cusp, the corresponding factor 
in the discriminant will occur as a factor of the third or higher odd-numbered de- 
gree, so that the cusp-locus does form a boundary between regions of a real and a 
imaginary, but it is easily distinguished from the envelope whose factor occurs in 
the first degree. In fact, the node locus and the envelope which would exist, if the 
members of the system were looped curves, may be regarded as coming into coin- 
cidence when they become cusped curves. 



35° APPLICATION TO PLANE CURVES. [Art. 360. 



which represents a parabola with axis vertically downward and 
focus at the origin. 

This envelope constitutes the boundary separating the 
points of the plane which can be hit by properly choosing a 
from those which cannot be reached. For points within it, 
there are two values of a, that is to say, two angles of eleva- 
tion, which can be used. For a point upon it, there is but a 
single value of a. 

General Method. 

361. Denoting by Aa the difference between two values 
of a, 

f[x, y, a) = o (1) 

and 

f(x, y, a -f- da) = (2) 

are the equations of two members of the system of curves, 
such as those passing through A and B, Fig. 69, p. 347, 
which can be brought as near as we please to coincidence by 
diminishing Aa. A point of intersection of these curves, such 
as P, Fig. 69, will satisfy both these equations, and therefore 
also their difference, 

f(x, y, a -f Aa) —f{x, y, a) = O. . . . (3) 

Regarding f(x, y, a) simply as a function of a, the first mem- 
ber of equation (3) is Af(x, y, a), which vanishes identically 
when we put Aa = o to obtain the ultimate intersection. But, 
if we divide by Aa, the ratio Af/Aa approaches a limiting 
value when Ax is diminished ; for, by Art. 40, we have 



)~| _ dfix, y 

J ia = da 



Af(%, y, a)~\ _ d/(Xy y, a) 

Aa 



§ XXIX.] GENERAL METHOD OF FINDING ENVELOPES. 351 
Hence equation (3) becomes at the limit 

df(x, y, a) 



d 



a 



= °> (4) 



and this equation is satisfied by that ultimate intersection 
which lies upon the curve (1), in other words, by the point of 
tangency of the curve (1) and the envelope. It follows that, 
if we eliminate a between equations (1) and (4), we shall have 
an equation satisfied by all the points of ultimate intersection. 
In other words, the equation of the envelope is the result of 
eliminating a between the equations 

f{x, y, a ) = o,\ 
f(x, y, a) = o, j w 

where f stands for the derivative off with respect to a. The 
result is called the discriminant of equation (1) with respect to a. 
362. For example, let us find the envelope of the system 
of ellipses whose axes are fixed in position, and whose semi- 
axes have a constant sum. Denoting the constant sum by c, 
the equation of the varying ellipse is 

x 2 v 

— 2 + 7 ^2 =1 ( l ) 

a 4 {c — <x) 



Taking the derivative with respect to a, 

2V 2 



2X 2 2 V 2 

7 + 7 ^3- — o; (2) 



whence 



x i y i 
a=c —j 1 and c — a = c -f 1 ; 

x 3 +y* x f -\-y^ 



35 2 APPLICATION TO PLANE CURVES. [Art. 362. 

and, substituting in equation (1), we find 

S 1 3 

(x*+y ) =c*. 

Therefore the envelope is the four-cusped hypocycloid, Art. 
292. 

Two Variable Parameters. 

363. When the given equation involves two parameters, 
a and /?, it represents a doubly-infinite system of curves, unless 
there is a given relation between the parameters (like that be- 
tween the semi-axes of the ellipse in the preceding example), so 
that one of them can be eliminated. Instead of eliminating 
one of the parameters at once, it is, however, sometimes 
preferable to proceed in the manner illustrated by the follow- 
ing example: 

The centre of a circle which passes through the origin 
moves upon the equilateral hyperbola 

x 2 — y 2 = a 2 ; 

required the envelope. Taking for the two parameters the 
coordinates of the centre, the equation of the circle is 

(x - a) 2 +(y- fi) 2 = oc 2 + /? 2 , 
or 

x 2 -f- y 2 — 2ax — 2/3y = O, . . . . (1) 

and the relation between the parameters is 

« 2 - /3 2 = a 2 . . ! . . . . (2) 



§ XXIX.] THE E VOLUTE AS AN ENVELOPE. 353 

The derivative equation, to be combined with these, is now 
found by eliminating the ratio d/3 : doc from the differential 
equations. Thus, from equations (i) and (2), we have 

xda= — ydfi and ada=fidfi\ 

whence 

xft = - ya (3) 

From equations (i) and (3), which are of the first degree with 
respect to a and ft, we find 

_ x(x 2 + f) _ y(x 2 + y 2 ) 

a ~ 2 {x 2 - y 2 ) ' P " " 2(x 2 -y 2 Y 

and substituting in equation (2), we have 

(x 2 + ff = 4a 2 (x 2 - y 2 ) 2 

for the equation of the envelope, which is therefore a lemnis- 
cate (see Art. 304). 

The Evolnte Regarded as an Envelope. 

364. We have seen (Art. 332) that the evolute of a curve 
is tangent to all the normals. Hence, if the equation of 
the normal to a given curve is expressed in terms of a single 
parameter, the evolute may be found as an envelope. 

For example, the coordinates of a point on the hyperbola 

x 2 y 2 _ 

J 2 ~V i=z *' 

when expressed in terms of an auxiliary variable, are 
x = a sec ip, y= b tan ip ; 



354 APPLICATION TO PLANE CURVES. [Art. 364. 

hence the equation of the normal at this point is 

a 
y —b tan tf> = — -sin tp(x — a sec 0), 



or 



by + a x s i n V ; = (a 2 + & 2 ) tan ^, (1) 



in which ^ serves as an arbitrary parameter for the system of 
normals. Taking the derivative with respect to ip f 

ax cos ip = (a 2 -f- 6 2 ) sec 2 ^, 

or 

a 2 + 6 2 Q , 

x — sec 3 ^ (2) 

a v ' 

Substituting" this value of x in equation (1), we find 

a 2 +b 2 , , 

y= g— tan ^ (3) 

Eliminating tp between equations (2) and (3), we have for the 
envelope 

(axf-(byf=(a* + b*f (4) 

Envelopes of Straight-line Systems. 

365. It should be noticed that, whenever the system is 
one of straight lines, each of the equations f(x> y, a) = o and 
f(x } y, a) = o is of the first degree with respect to x and y ; 
and therefore it will always be possible to express x and y in 
terms of a as a third variable, although the final elimination 
giving the rectangular equation may not be practicable. 



XXIX.] ENVELOPES OF STRAIGHT-LINE SYSTEMS. 355 



For example, let us find the envelope of the reflected rays 
when parallel rays of light fall on a y« 

semicircular mirror. Let AP, Fig. __ 
70, a ray parallel to the axis of x, a 
meet the circle at P, where the an- 
gle POR at the centre is a. Then, 
because the angles of incidence and 
reflection APO and OPR are equal, 
the inclination of PR to the axis of x is 
2a. It follows that the intercept OR 
= \a sec a. Hence the equation of 
the reflected ray may be written 

y cot 2a=z x — \a sec a. 

The derivative with respect to a is 




Fig. 70. 



W 



2y 



a sin a 



sm*2a 



\2) 



2 cos'a 
whence 

y=as'm 3 a, (3) 

and substituting in equation (1), 

x = \a cos <*(3 — 2 cosV) (4) 

Comparing these equations with Art. .282, we see that the 
envelope is the two-cusped epicycloid in which the radius of 
the fixed circle is \a and that of the rolling circle is \a. 

Envelopes of this character are called caustics, and a cusp 
like that at (ja, o), the limiting position of R in Fig. 70, is called 
the focus (burning point) because a large number of reflected 
rays pass very nearly through it. 

366 8 Another example of the envelope of a system of 
straight lines is the curve of which a given curve is the pedal- 
In Fig. 66, p. 330, if the curve BPC is given, the locus 



356 APPLICATION TO PLANE CURVES. [Art. 366. 

DRE of R, the foot of the perpendicular upon the tangent 
from a fixed point 0, is called the pedal of BPC Conversely, 
when DRE is given, BPC is called its negative pedal with respect 
to the pole O. Thus the negative pedal is the envelope of the 
perpendicular to the radius vector OR at its extremity. 

Now using r and to denote the polar coordinates of the 
point R on the given curve, the equation of the perpendicular 
will be 

x cos + y sin 6 = r> (1) 

in which we now regard as the arbitrary parameter, while r 
is a known function of 6. Taking the derivative, we have 

dr 
— x sin 6 + y cos = — , . , . . (2) 

and eliminating y and x successively, we obtain 



dr . 
x = r cos 6 — -,- sin 0, 
ad 

dr 
y = r sin + -=- cos 0, 
at) 



(3) 



the rectangular coordinates of the negative pedal. 

367. For example, let it be required to determine the nega- 
tive pedal of the strophoid, the node being the pedal origin. 

The polar equation of the strophoid referred to its node, 
found by transforming equation (1), Art. 257 (and reversing 
the direction of the initial line), is 

a cos 26 , 

r = — -p~ = a (cos a — sm u tan 0) ; 

cos 6 

whence 

dr 

— = — a sin 0(2 -f- sec 2 #). 



§ XXIX.] examples. 357 

Substituting in equations (3) of the preceding article, and 
reducing, we have 

x = a sec 2 # and y = — 2a tan B\ 

whence, eliminating 6, 

y 2 = 4a(x — a). 

Hence the negative pedal is a parabola whose vertex is 
situated at the point (a, o), the vertex of the strophoid. , 

Examples XXIX. 

1. Find ths envelope of the system of parabolas represented by 
the equation 

y = 7 (* - «)» 

in which a is an arbitrary parameter and c a fixed constant. 

y 2 = — x z . 

2. Find the envelope of the circles described on the double or- 
dinates of an ellipse as diameters. x 2 y 2 

a 2 + b 2 + W~~ lm 

3. Find the envelope of the ellipses, the product of whose semi- 
axes is equal to the constant c 2 . 

The conjugate hyperbolas, 2xy = ± c 2 - 

4. Find the envelope of a perpendicular to the normal to the pa- 
rabola, y 2 = 4<zx, drawn through the intersection of the normal with 
the axis. y 2 = 4^(2*2 — x). 

5. Show that the hyperbolas given by the equation 

a 6 

1 — = 1, when a -\- 6 = c, 

x y 

form a pencil, and therefore do not admit of an envelope. 

6. A circle moves with its centre on a parabola whose equation is 
y 2 = ^ax, and passes through the vertex of the parabola ; find the 

envelope. The cissoid, y\x -j- 2d) -j- x z = o. 



358 APPLICATION TO PLANE CURVES. [Ex. XXIX. 

7. A straight line cuts the coordinate axes in such a manner that 
the product of the intercepts is constant and equal to c 2 ; find the 
envelope. xy = \c 2 . 

8. A perpendicular to the tangent to a parabola is drawn at the 
point where the tangent cuts the fixed line x = c; find the equation 
of the envelope of this perpendicular. 

The parabola, y 2 = — 4c(x — a — c). 

9. Circles are described upon the double ordinates of the parabola 
y 2 = 4.ax; determine what portion of them fail to touch the envelope 
of the system. 



a 

10. Find the curve to which y =mx A is tangent. 

m 



y 2 = 4 ax. 

11. The centre of an hyperbola passing through the origin, and 
having asymptotes parallel to the axes, moves upon the circle 
x 2 -\-y 2 = a 1 ', find the envelope. x 2 y 2 = a 2 (x 2 -j- y 2 ). 

12. Find the envelope of the parabolas 

X s y 2 

-T + -7 = I 

(which touch the coordinate axes at the distances a and fi from the 
origin), when a (5 = c l . i6xy = c 2 . 

13. The intercepts a and (3 of a straight line on any two coordi- 
nate axes are connected by the linear relation 

na-\- (3 = c ; 
find the envelope. 

The parabola, (y — nx) 2 — 2ticx — 2cy -f- c 2 = o. 

14. Find the envelope of the system of curves, 

x n v n 
when 



j£*n-\-n _1_ ym-\-n — - £tn-\-n 



§ XXIX.] examples. 359 

15. From a point in the ellipse perpendiculars are drawn to the 
axes ; find the envelope of the line joining the feet of these perpen- 
diculars. ^"_i_ (Z\* -- 



16. A straight line of fixed length a moves with its extremities on 
the two rectangular coordinate axes; determine the envelope. 



2 2 



xs -J-^3 — a 



17. Let a perpendicular to the line AB in example 16 be - drawn 
through the extremity which slides in the axis of x ; find the en- 
velope, and prove that it is one of the involutes of the astroid whose 
semi-axis is ia. x = a sin 2 <z cos a -\- a cos or, 



y = a sin a cos 2 *?. 

18. Find the envelope of the circles whose centres are on the 
fixed circle x 2 -\-y 2 = a 2 and which touch the axis of x. 

The two-cusped epicycloid \{x 2 -j- y 2 — a 2 ) s = 2']a x y 2 , 

19. Find the equation of the evolute of the parabola y 2 =4ax, 
using the equation of the normal in terms of its direction-ratio, viz., 

y = mx — 2am — am 3 . 

2jay 2 = 4(x — 2af. 

20. Find the equation of the evolute of the cycloid, by means of 
the equation of the normal in terms of ip. 

The equation of the normal is 

sin tp 

x 4- v r —aty = o. 

1 — cos tp r 

The equations of the evolute are 

x = a(ip -J- sin tp) and y = — a(i — cos tji). 

Compare Art. 339. 

21. From any point C on the circumference of a circle whose 
radius is a an ordinate to the fixed diameter AB is drawn, and 
through the foot of the ordinate a perpendicular to the chord AC is 
drawn ; find the equations of the envelope of this perpendicular, and 
trace the curve. 



360 APPLICATION TO PLANE CURVES. [Ex. XXIX. 

Denoting by 6 the angle BAC, and taking the origin at A, we 

derive x — 2a cos 2 6(s — 2 cos 2 #), ) 

y = — 4a sin cos 3 #. j 

The curve is symmetrical to the axis of x. 6 = gives the point B> 
= 30 gives a cusp, and 6 = 90 a cusp at the origin. 

22. In the figure of Ex. 21, the foot of the ordinate is joined to the 
middle point of AC; find the envelope, and show that it is the same 
as the result of Ex. 21, with A and B interchanged. 

x = 2a cos 2 # cos 26 ' ) 
y = 2<2 sm 2 6 sin 26. j 

7%e c#rz>£ z> //fo three-cusped hypocycloid. 

23. Determine the negative pedal (see Art. 366) of the parabola 
y 2 = ^ax with respect to its vertex. 

The semicubical parabola (x — 4a) 3 = 2 'jay 2 . 

24. Determine the negative pedal of the cissoid 

sin 2 # 

r = 2a 

cos u 

y 2 = — 8ax. 

25. Prove that the negative pedal of the spiral of Archimedes is 
the involute of the circle. 

26. Show that the negative pedal of the curve r z= b sin md is de- 
termined by 

x = b sin md cos 6 — mb cos md sin 6, 
y = b sin md sin -\- mb cos md cos #, 

being a hypocycloid or an epicycloid, according as m is greater or 
less than unity. 

27. Find the caustic of the circle when the incident rays pro- 
ceed from a point on the circumference. 



The cardioid x = \a cos 26 — ^a cos 46, 
y = |# sin 26 — \a sin 46 



':} 



§ XXIX.] EXAMPLES. 361 

28. Derive the polar equation of the negative pedal of the curve 

r m = a™ cos md. 
The rectangular equations are 

T — tn z—m 

x = a(cos md) ** cos (1 — m)6, y = #(cos md) m sin(i —m)6. 
Whence, denoting the polar coordinates of the pedal by r' and 6' , 

tan 6' = — = tan (1 — m)6 i and r'= a(cosm6)~^T. 

Therefore, eliminating 6, 

, — - md' 

r 'x-m = a z-m CQS m 

i—m 



CHAPTER VIII. 
Functions of Two or More Variables. 



XXX. 



Partial Differentials, 

368. When u denotes a function of the single variable x, 
the differentials which measure at any instant the rates of 
variation of x and u are connected by the equation 

du =f'(x)dx } 

where f'(x) is the derivative of u with respect to x. Accord- 

du 
ingly. when -=— is used as the symbol of this derivative, du in 

the numerator is the measure of the rate of u due to the varia- 
tion of x. Now if u is also a function of v, the du in the 

du 
symbol -=- for the derivative with respect to y is a new quan- 
tity, having a like relation to the variation of y. These two 
quantities, which we may distinguish at present by the sym- 
bols d x u and d y u, are called partial differentials of u, while du, 
which measures the actual rate of u when both x and y vary, 
is called the total differential of u. 

369. We have seen in Art. 86 that, as a consequence of 
the rules of differentiation, the total differential of any func- 



§ XXX.] PARTIAL DIFFERENTIALS. 363 

tion expressible in terms of the elementary functions is the 
sum of the partial differentials; or, in the present notation, 

du = d x u -\- d y u. 

We shall now show that this is true for all continuous func- 
tions, as a consequence of the relation between differentials 
and finite differences. 
Let 

u=f(pc,y), (1) 

u' = f{x + Ax, y), (2) 

u"=A*+**>y+*y)\ • • • • (3) 

then if A x u denotes the increment due to increasing x by Ax, 
and a like meaning is given to A y u, while Au indicates the 
increment due to an increase in each variable, we have 



A x u- 


= u' 


— u, 


Am' = u" — u, 


nce 






Au — A x u -\- A y u' 



Au = u" — u, 



(4) 



Dividing each member by At, the increment of time corre- 
sponding to Ax and Ay, we have 

Au Am A v u r 

+ -5T- ..... (5) 



At At ' At 



Now, by Art. 24, each of these ratios has for its limit the 
corresponding rate; in other words, it differs from the rate 
by a quantity e which vanishes with At. Thus we may write 



du d x u f d y u r 

di + 6:=Z ~df + 6 ' ~dt 



-£ + e = ^+e'+^+e", . . . (6) 



364 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 369. 

in which e, e r and e" vanish with At But, when At — o, 
Ax = o and, by equation (2), u f = u; hence, when At = o, 
equation (6) becomes 

di = ~df + ~df ( 7) 

Therefore 

du = d x u -f- J^w (8) 

This principle is readily extended, by the same method, to 
functions of several independent variables. 

Partial Derivatives. 

370. The theorem of partial differentials is sometimes 

written 

du du 
du = Tx dx + jjdy, (1) 

in which the coefficients of dx and dy (which are the deriva- 
tives of u with respect to x and to y) are called partial de- 
rivatives. Their numerators, though identical in form, are, 
as shown in Art. 368, really equal to d x u and d y u respect- 
ively. The absence of distinctive marks will however pro- 
duce no confusion, if it is understood that the fractional form 

du 

— will be used only to express the derivative of u with respect 

ax 

to x, no matter how many independent variables may be 
under consideration. 

371. If x and y are made functions of any other variable 
t, u becomes a function of /, and equation (1) gives 

du du dx du dy 

dt = dx"dt + dy dt' (2) 



§ XXX.] PARTIAL DERIVATIVES. 365 

in which the derivative of u with respect to / is expressed in 
terms of the partial derivatives due to its expression as a 
function of the two independent variables* x and y. 

Geometrical Representation of Partial Derivatives. 

ZIIm Let the independent variables^ and y be rectangu- 
lar coordinates of a point R in the horizontal plane, and let u 
be the vertical third coordinate of a point P in space; then 

w=f(x, y) (I) 

is the equation of a surface. If a plane parallel to the plane 
of xz be passed through PR, y will have a constant value in 
this plane; hence when y is regarded as a constant, equation 
(1) will become the equation of the intersection of this plane 
with the surface. It follows that if xp Y is the inclination to 
the plane of xy of the tangent at P to this curve, tan ip x will 
represent the derivative of u with respect to x. Denoting, in 
like manner, by ip 2 the inclination of a line tangent to the 
section of the surface made by a plane parallel to that of yz, 
we have 

du du 

tan * = dx' tan *■ = dy ■ 

* If x is identical with the new variable / (which is as much as to suppose that 
y is given as an explicit function of x) t the equation becomes 

(du\ du du dy 

\dx) dx ' dy dx' 

in which a special mark must be used to distinguish the total derivative of u with 
respect to x (after y is eliminated) from the partial derivative in which y is re- 
garded as independent. This distinction is sometimes made by adopting a differ- 
ent form of the letter d in the partial derivatives ; thus 

du du du dy 
dx dx dy dx ' 



366 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 373. 

373. At all ordinary points of the surface, the plane which 
passes through these two tangent lines is a tangent plane 
to the surface. Now, when R moves in any manner due to 
the variation of x and y, P moves in some curve drawn in the 
surface, and the tangent to its path lies in this tangent plane. 
It follows, as in Art. 37 (see Fig. 7, p. 27), that the incre- 
ment of u, as measured up to this tangent plane, represents the 
hypothetical increment which is denoted by du in accordance 
with Art. 23. 

If ds is the differential of the actual motion of R in the 
horizontal plane, the total differential du corresponds to the 
motion ds, and the partial differentials d x u and d } u corre- 
spond to the resolved parts of this motion in the directions of 
the axes. The w-coordinates of the tangent plane, corre- 
sponding to the four corners of the rectangle whose sides are. 
dx and dy, are u, u -\- d x u, u -f- d y u, and u -f- du. These form 
the edges of a truncated prism, and it is readily proved geo- 
metrically that the sums of opposite edges are equal. Hence 

du = d x u -j- djU. 

du 

374. The derivative -=- =: tan *p f where is the inclina- 

ds 

tion to the horizontal plane of the tangent to the actual path 
of P. It is called a " total derivative" only with reference 
to the partial derivative theorem (equation (2), Art. 371), 
which gives its value in terms of derivatives with respect to 
x and y, namely, 

du du dx du dy 

ds dx ds dy ds J 

If <f> denotes, as usual, the inclination of ds in the plane of xy 
to the axis of x, equation (1) becomes 

du du du 

J7 = ^ COS ^ + jy sm * 



§ XXX.] HIGHER DERIVA TIVES. 367 

This is, in effect, a relation between the trigonometric tan- 
gents of the inclinations to the horizontal plane of the three 
tangent lines in the surface ; that is to say of the angles we 
have denoted */>, ip x and ^ 2 .* 

Higher Derivatives. 

375. As in Art. 97, we may regard -=- and -=— as symbols 

of the operation of taking the derivative, which may be 

detached from the operand u. Single letters may also be 

used as in Art. 10 1, a distinctive mark being employed, in 

this case, to indicate the independent variable. Thus we 

_ d d 

may put D = -p and D = -*-. 

The partial derivative of u with respect to either variable 
will in general be a function of both variables, and will there- 
fore admit of a derivative with respect to either variable. 
These derivatives are called the derivatives of u of the second 
order. Thus we have the four derivatives 

d du d du d du d du 

dxdx dydx y dxdy' dy dy' 

Using the abbreviated notation, introduced in Art. 97 in the 
case of a single independent variable, these are usually 
written 

d 2 u <Pu d 2 u d 2 u 

dx z ' dydx' dxdy dy 2y 

in which no separate meaning has been given to the numer- 
ators, and the order of the operations is that of the factors in 
the denominator. 



* This relation may be verified by spherical trigonometry. 



3^8 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 375. 



Again, they may be written in the symbolic form 
D 2 u, D'Du, DD'u, D f2 u. 

376. If we use A to denote the operation of taking the 

finite or actual difference between two values of a function, 

consequent upon an increment of one variable, we may use 

A A 

— and — for the operation of taking such a difference and 

Ax Ay r 

then dividing by the increment upon which it depends. 

Then, corresponding to every derivative of the second or a 

higher order, we have a corresponding combination of these 

symbols. We shall now show that each of the higher 

derivatives is the limit of the value of the corresponding 

zf-symbol when Ax and Ay simultaneously vanish. 

377. For this purpose, let us resume the consideration of 
the equation of Art. 40, 

Au du , N 

Tx=Ji+ e ' (1) 

in connection with which this relation is expressed, for the 
first derivative, by the statement that e vanishes with Ax. 
Now this quantity e is a function not only of Ax, but of the 
independent variables x and y. Hence it admits of differen- 
tiation with respect to each of these variables. But, since e 
always assumes the value zero when Ax = o, it is, for this 
value of Ax, independent of x and y. Therefore the derivatives 

de de 

-7- and -7- vanish with Ax. 

ax ay 

378. Equation (1) shows that, in order to apply the symbol 

— — to a function of x, we must change A to d and then add a 
Ax fa 



§ XXX.] HIGHER DERIVATIVES. 369 

quantity which vanishes with Ax. Thus, applying the opera- 
tion to equation (1) itself, we have 



A Au _ A /du \ 
Ax Ax Ax\dx ■ J 



d /du 
dx 



_ d z u de f 
= dx? + dx + 6 ' 

de 
in which both — and e f vanish with Ax; so that we may write 

A 2 u _ d 2 u 
Again, applying the operation — to equation (1), we find 



A Au d /du \ 
Ay Ax ~~ dy\dx ~^~ J " 1 ~ 



e" 



+ ¥ + e", 



dy dx dy 

de 
in which -=- vanishes with Ax and e" vanishes with Ay; so 

that we may write 

A 2 u d z u 



Ay Ax dy dx 



+ e (3) 



where e vanishes when both Ax and Ay vanish. 

In like manner, it may be shown that the ^-symbols of 
higher orders, whether there are two or more independent 
variables concerned, have for their limits the corresponding 
higher derivatives. 



37° FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 379. 

Commutative Character of Differentiation. 

379. We have hitherto attached no meaning to the sep- 
arate terms of the symbols in fractional form employed in the 
preceding articles. But with respect to the ^/-symbols this is 
readily done. Thus, in equation (2), the first member is the 

Au 
ratio to Ax of A——. This last denotes the increment of a 

Ax 

fraction whose denominator is constant, which is obviously 
the same as the result of dividing the increment of the numer- 
ator by the denominator; that is, 

Au A.Au 

Ax " Ax 

Thus A 2 u in equation (2) is simply the abbreviated symbol 
for the increment of Au due to the increment Ax in the inde- 
pendent variable x. 

380. With respect to the symbol A 2 u in equations (2) and 
(3), a distinction must be made between increments due to 
Ax and to Ay respectively. Using suffixes for this purpose, 
the A 2 u in equation (3) is an abbreviated form of A y (A x u). 
Now, putting 

u =f(x,y), 

we have, as in Art. 369, 

A x u =f(x + Ax, y) - f{x, y) ; 

if in this equation we replace y by y -f- Ay, we obtain a new 
value of A x u, and, denoting this value by A' x u, we have 

A' x u =f(x +Ax,y+ Ay) —fix, y + Ay). 



§ XXX.] DIFFERENTIATION COMMUTATIVE. 371 

Since this change in the value of A x u results from the incre- 
ment received by y, the increment received by A x u will be 
A y (A x u)\ hence 

A y {A x u) = A' x u —A x u, 
or 

A y (A x u)=f{x-\-Ax, y+Ay)—f(x, y+Ay)-f(x+Ax, y)-{-f{x,y). 

381. The value of A x (A y u)> obtained in a precisely similar 
manner, is identical with that just given ; hence 

A y {A x u) = A x {A y u) (i) 

It follows that the values of the fractions 

A y A x u A x A y u 

Ay Ax Ax Ay 

are the same ; whence we infer that the derivatives which are 
their limiting values when Ax and Ay vanish are also equal. 
Thus 

d 2 u d 2 u 

dy dx dx dy ' 

The theorem expressed by this equation is readily verified in 
any particular case. Thus, given 

u = y x , 
we have 

du du 

taking the derivative of the first with respect to y, and that of 



37 2 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 38 1. 

the second with respect to x, we obtain the same expression, 
namely, 

y x ~ I (x log y + 0- 

382. The result arrived at above may be expressed thus: 
the operations of taking the derivative with respect to two inde- 
pendent variables are commutative; that is, they may be inter- 
changed without affecting the result obtained. 

This theorem may be extended to derivatives higher than 
the second, and also to functions of more than two inde- 
pendent variables. For it has been proved that we may, 
without affecting the result, interchange any two consecutive 
differentiations, and it is obvious that, by successive inter- 
changes of consecutive differentiations, we can alter the order 
of differentiation in any manner desired. Hence all differen- 
tiations with respect to independent variables are commu- 
tative. 

Accordingly, the result of differentiating m times with re- 
spect to x, n times with respect to y, and p times with respect 
to z, may, without regard to the order of differentiation, be 
expressed by the symbol 

dx m dy"dz p 

Commutative and Distributive Operations. 

383. When m is constant we have 

d (mu) = mdu ; 
hence when u is a function of x. 

d d ■ 

— mu = m^-u (i) 

dx dx v ' 



XXX. j DISTRIBUTIVE OPERATIONS. 373 



This equation indicates that the operation of multiplying by 

a constant and the operation of taking the derivative with 

reference to x are commutative. It follows that the factors 

d 
of the symbolic product y-ir, when y is a variable independent 

of x, are commutative; for, in performing the operation -=-, y 

is regarded as a constant. 

d 

On the other hand, x-=- is not commutative ; for 

ax 



du 

~Z JvU J\ 

ax 
while 



, XU — X 7 H U% 

ax ax 



d du 

3v ~~ ^ U 00 ~^ a 

ax ax 

384. A repeated application of the same symbol, whether 
simple or compound, is indicated by affixing an index to the 
symbol ; thus, 

<A 2 d d 



y dx/ ^ dx y dx' 

and, since the operations indicated by the symbol are com 
mutative, we have 

d V . d 2 



ydx) = y 2 dx^- •'•-■■• (o 

On the other hand, since the operations indicated by the 

d . / A 2 . , .<? 

symbol x-r- are not commutative, I x-r~ 1 is not equal to ^-r^- 

We have, in fact, 

d V du . . d?u 

*si) u = *&+*■& (2) 



374 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 385. 

385. The result obtained by adding the results of the 
application of two operative symbols is expressed by prefixing 
to the operand the sum of the operative symbols. For exam- 
ple, the result written above may be expressed thus : 

\ dx] " dx dx 2 ' 

for this is a general formula applicable to any function of x as 
operand. 

On the other hand, we may write formulae involving 
special forms of the operand, thus 

(x£)x" = MX-, 

whence it readily follows that 




386. An operation which, when applied to a sum, gives 
the sum of the results of separate application to the parts is 
said to be distributive over a sum. Thus, since 

d d d 

dx( U + v) = Tx U + dx V > 

differentiation is a distributive operation. In this formula, 
the symbol of differentiation is applied to a sum exactly as 
if it were an ordinary algebraic factor; in like manner, every 
combination of differential symbols and algebraic multipliers 
has the same property. 



§ XXX.] SYMBOLIC IDENTITIES. 375 

In algebra, an exponent is distributive over the factors of 
a product, as expressed by the formula (abc) m = a m b m c m . But 
this is not true of operative symbols unless the symbolic factors 
are commutative. Compare equations (i) and (2), Art, 384. 

Symbolic Identities. 

387. The formulae of algebraic expansion are consequences 
of the commutative and distributive nature of algebraic multi- 
plication ; hence it follows that a symbolic product or power 
may be expanded by these formulae ; provided all the con- 
stituent symbols represent commutative operations. Thus 

id \ld \ d 2 u 

Again, when the constituent symbols are not commutative, 
we may obtain symbolic identities by the rules of differentia- 
tion. Thus, let u denote a function of d; then we have, by 
differentiation, 

d ( d \ 

Td endu=e \d0 +n r 

and multiplying by e~ nd , 

d I d \ 

e ~ ne Te end ' u = \dd + n ) u (I) 

Applying now the symbols whose equivalence is expressed in 
this equation to the equation itself, we have 

d d d 2 Id \ 2 



376 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 387. 

and, by repeating this process, we have in general 



dO' 6 ' u -\dd 



— r e nd >u=[-j; + n)-u* .... (2) 



Eulers Theorems concerning Homogeneous Functions. 

388. A homogeneous algebraic function of the wth degree 
involving two variables may be put in the form 

*=*'($) « 

Any expression which comes under this form, even when f \s 
a transcendental function, and when n has a fractional or 
negative value, is called a homogeneous function. Thus 

_ Vx-\r Vy 
x{x -\- y) 

is a homogeneous function in which n = — f . Again, 

u = log [x -j- Vip^ + /)] — l°g x 

is a homogeneous function in which n — o. 

389. By differentiating equation (1), we derive 

l=«-^)-.<- V (i). 

and 

' *£■= *-*/>(! 

dy \x 

* This equation is equivalent to the result obtained by means of Leibnitz' 
theorem in Art. 107. 



XXX.] EULER'S THEOREMS. S77 



whence 

du du 

v dx ' ^ dy 
or, symbolically, 



du J y\ 



( d d\ 

[ x di+y^) u=m & 

when u is a homogeneous function of the wth degree. 

Again, the derivatives of u are homogeneous functions of 
the (n — i)th degree; hence, by the theorem expressed in 
equation (2), we have 

id d\du . du / d , d\du du 

\ X dx + y Ty!Tx = {n ~ ! te and { X Tx + yTy)Ty = ( "~ l) Ty ; 

whence, expanding, 

d 2 u <Pu , .du d 2 u d 2 u du 

x dx> + ydyT*= in ~ l) Tx' and X JxTy + y^ ={n - l) Tf 

multiplying by x and y respectively and adding, 

9 d 2 u ' (Pu A 2 u . r du du~ 

*d* +2Xy alay + y df = {n ~ ^Tx + ydy-j ' 
hence 

d 2 u d 2 u d 2 u 

x ^ + 2X yixTy + ydy = n(n - I)u - ■ ■ (3 ) 

The results expressed in equations (2) and (3), and similar 
results involving higher derivatives, are known as Euler's 
Theorems concerning homogeneous functions. 



37$ FUNCTIONS OF TWO OR MORE VARIABLES. [Ex. XXX. 

Examples XXX. 

i. Given u = log [x + \/(x 2 +y j )], prove that 

/ d d\ 

2. Given u = log (~v 3 + j ;3 -f- s 3 ~ 3 x y z )> prove that 

</« du du 3 

j„ ~T~ ~r~, ~r "T~ 



<a?jt: ^ dz x -\-y -j- 2 

3. Given « = sec (y -f- &#) -j- tan ( y — a*), prove that 

d 2 u d 2 u 

dx' 1 dy 2 ' 

d u d 2 u 

4. Verify the theorem = when w = sin (jry 2 ). 

5. Verify the theorem = when « = log tan (ax -\-y 2 ). 

d u d u jc 

6. Verify the theorem . ., . - = -= — rs when « = tan -1 —. 

J ay 4 dx dx dy 4, y 

d^u d^u 

7. Verify the theorem j-^ = ^^ when u =y\og (1 + ^). 

8. Given « = sin x cosy, prove that 

d*u d*u d*u 



dy 2 dx 2 dx 2 dy 2 dxdydxdy 

1 

q. Given u = ■— — = -, prove that 

y +/(4ao — c 2 ) 

d 2 _ d 2 
dc 2 da db 

10. Given u = (x -\-y) 2 , prove that 

d*u a 2 u du 

dx 2 dxdy dx' 



XXX.] examples. 379 



ii. Given u— -, prove that 

(x 2 -\-y 2 -{-z 2 y 

d 2 u d 2 u d 2 u 

dx 2 + df "*~ ~d? == °* 

12. Given u = log (jf 3 +jy 3 -\-z z — 3^^), prove that 
cPu d 2 u d 2 u d 2 u d 2 u d 2 u o 

+ TT> + l-o + 2 ^—7, + 2 — - + 2 



dlx 2 dy 2 dz 2 dxdy dy dz dzdx {x -\-y -j- z) 2 . 

/</ </ <A 2 
Employ the symbol ( - — [- i" + ^-) w > # w ^ J ^ ^£k. 2. 



Jtrj/ 



13. Verify Euler's theorems for u = {x 2 -\-^y y and for u = 

14. If the rectangular axes of x and y are turned through the 

angle in the case of a surface z = /~(x ) y), find-—, and ~ in terms 

° r dx dy 

of — and — : and thence show that —7- may be regarded as a " total 
^ dy' dx 

derivative ' ' when -7-7 and -r-. are the partial derivatives. 
dx dy 

15. Prove the symbolic equation 

d m ^ d* . d m ^ 

-, x m+p — . • u = x p — — -j. x m • u. 



dx m dx* dx m+p ' 

by showing the identity of the result of each operation upon x r , and 
therefore upon any function developable in a series of powers of x. 

16. Apply the symbol a Q ~ V- 2a. - — \--\a^—- to the expression 

da x 1 da z 2 da % 

a Q \ 2 - da Q a x a % a % + 4 a a* + 4<z*a s - 3 a 2 a 2 . 

Result o. 



380 FUNCTIONS OF TWO OR MORE VARIABLES. [Ex. XXX. 

1 7. Operate on <vy* 4 + 2a x a 2 a z ~ a o a z — a ? a K — a ? with the symbol 
d d d d 

°da, l da„~ * 2 da. ~ z da/ 



Result o. 



18. Determine the value of 



du du x 2 — y 2 

x — — \-y-r, when u = tan -1 — . 

ax ay ax 

Solution : 

Since tan u is a homogeneous function of the first degree, we have, 
by Euler's theorem, 



/ du , du\ 



dy 
whence 

du du ax(x 2 —y 2 ) 

X lx"^ y ~ay = a*x*+(x* -y 2 ) 2 ' 

19. If u = sin z;, z> being a homogeneous function of the «th de- 

. . . r du du 

gree, determine the value of x ~-\-y — . 

c/*v CI Z4> 

du du 
x ~r -\-y -r = nv cos v. 
dx dy 



XXXI. 

Change of the Independent Variable. 

390. It is frequently desirable to transform expressions 
involving derivatives with reference to x into equivalent ex- 
pressions, in which some variable connected with x by a 
known relation is the independent variable. This process is 
called changing the independent variable. 



§ XXXI.] CHANGE OF THE INDEPENDENT VARIABLE. 38 1 

Let y denote the function whose derivatives occur in the 
given expression, and let 6 denote the new independent vari- 
able. For the first derivative, we have 

dy _ dy dd 
dx dd dx' 

dd . 
in which it is necessary to express -=- in terms of d by means 

clx 

of the given relation. For example, suppose this relation to 

be 

x = tan d; . (1) 

differentiating, dx = sec 2 6 dd, whence, 

dd 

fa = cosd > • ( 2 ) 

and we have, in this case, 

%=%c»*0 (3) 

dx dd KJJ 

391. To express the second derivative with respect to x 
in terms of d, it is necessary to differentiate this expression 
as a product of variable factors. Thus 



d 2 y d Vdy 

— = — — cos 2 # 

dx 2 ddldd 



dd 
dx 



r d 2 y dy~\ 

= cos 2 cos 2 d-zj 2 — 2 cos d sin d ~ \. . (4) 

In like manner, expressions for the third and higher deriv- 
atives may be obtained. 



382 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 39 1. 

It will be noticed that, while the result in the case of the 
first derivative might be obtained by substituting for dx in 
the denominator its value in terms of 6, the second deriva- 
tive cannot be found in a similar manner. This is due to the 
fact that a common meaning cannot be assigned to the sym- 
bols ' d 2 y ' which constitute the numerators of the second 
derivatives. Compare the expressions of which the second 
derivatives are the limits, Art. 379. 

392. Change of independent variable may be used to 
simplify differential equations. For example, the differential 
equation 

d 2 y n dy 

( i + x *fd£+ 2x (* + x )£+ y = ° 

is transformed by means of equations (i), (3) and (4) above 
into the much simpler form 

d 2 y 



Transformations Involving Partial Derivatives. 

393. Let u denote a function of x and v, and let r and & 
be two new independent variables connected with x and y by 
two given equations, by virtue of which x and y are functions 
of r and 6. Then u is also a function of r and 6, and it may 
be required to express the partial derivatives of u with refer- 
ence to x and y in terms of derivatives with reference to r 
and 6. 

In Art. 371, the two independent variables x and y are 
supposed to be made functions of a new variable t, and equa- 
tion (2) shows how to obtain the derivative with respect to 



XXXI.] TRANSFORMATION OF PARTIAL DERIVATIVES. 383 



du 
this new variable. In finding the required expression for — , 

0/X 

r and are the two independent variables, and x takes the 
place of t) thus we have 

du du dr du dd 

dx~ dr dx ■ d0 dx' ^ ' 

, . , dr ' d . , 

in which -7- and -7- are to be derived from the given relations 
dx dx & 

between the two pairs of independent variables, just as -=-, in 

U/X 

Art. 390, is derived from the single relation there given. 
Similarly we have 

du du dr , du dd 

— = . (2) 

dy dr dy dd dy' v ' 

- dr dd 

in which -7- and -=-' are found in like manner. 
dy dy 

The four coefficients are, in fact, the derivatives of r and 
when x and y are associated together as a pair of independ- 
ent variables, but their values are to be expressed in terms of 
r and 6. 

394-. For example, let us assume that the given relations 
are those which connect polar and rectangular coordinates, 
namely, 

x = r cos 6 and y=rsmd. . . (1) 

Solving these equations for r and 6, we derive 

y 

r 2 = x 2 4- y* and = tan -1 - ; . . (2) 



384 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 394. 



whence 
dr 



x 



dx \/{x 2 + y 2 ) 



= cos#, 



dd 

dx 



sin d 



dr y _ . dd 

Ty ~ VWT7)~ Sm dy 



x 2 ~\- y 2 



x 



* 



cos 6 



}-. (3) 



x 2 + y 



Substituting these values in the general equations of the 
preceding article, we have 



du _ du du sin 6 

dx dr dd r ' 

dw dw dw cos 6 

dy dr do r 



(4) 

(5) 



Partial Derivatives of the Second Order. 

395. In finding expressions for the derivatives of the 

1 1 , 1 d u , du , . , 

second order, it must be remembered that -7- and -rr, which 

dr dd 

occur in the second members of equations (1) and (2), Art. 

393, are themselves functions of r and 6. Thus, by equation 

(2), Art. 371, 

d (du\\ d 2 udr d 2 u dd 

dx \dr / ' dr 2 dx dr dd dx' 

* It should be noticed that these values cannot be found by differentiating 
equations (1) with respect to r and d; but — and — could be found by elimina- 
tion from the partial derivatives of these equations with respect to x, which are 



and 



a dr . dd 

1 == cos -. r sm 6 -y 

dx dx 



dr ad 

o = sin 6 -r 4- r cos 6 -7-. 

dx dx 



d 2 u 



■j- This expression should not be written - — —, because that would indicate a 



dx dr 



§ XXXI.] DERIVATIVES OF THE SECOND ORDER. 



385 



In like manner, we have 

d /du\ 
dx\dd) "" drdddx~~lid 2 dx 



d /du\ d 2 u dr d 2 u dd 
~ 1 ~Jffz 1Z' 



Now, making use of these results, we derive from equation 
(1), Art. 393, ■ 



d 2 u 

dx 



du(Pr 
dr dx 2 



dr 



i£ ,■-/.** y» /\/*£ ri /\a 



'd 2 udr d?u dd' 



dx\_dr 2 dx ' drdddx. 



dud 2 d 
d6dx : 



~T~ ja j^2 \ 



ddF d?u dr_ (Puddl 
dx[_dr dd~dx~^"dd 2 dxA 



or 



d 2 u du d 2 r du d 2 6 
dx 2 ™ dr dx 2 ' dd dx 2 

(Pu/dr^y d 2 u dr_dd d?u/dd\ 2 

~^~ d^Kdx) +2 drd6dxdx^~ dd^Xdx) ' 



d 2 u 



d 2 u 



dx dy dy 2 



In like manner, expressions for the derivatives 

can be found. 

396. In applying this general expression to a special case, 

, ,. . 1 «. . d ' r , d 2 6 fM dr f dd 

the additional coefficients -7— 9 and ^—5 are, like -7- and -=-. to 

dx 2, dx 1 dx dx 

be derived from the given relations between the pairs of inde- 
pendent variables. For example, when the relations are 
those given in Art. 394, we derive, from the first two of the 
expressions included in equations (3), 



d 7 r 

dx 2 



d_ 
dx 



x 



Y 



(x 2 + y 2 y (x 2 + y 2 )i 



sin 2 fl 
r 



partial derivative of u when expressed as a function of r and x regarded as two 
independent variables. 



386 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 396. 

and 

d 2 6 _ d — y 2xy _ 2 sin 6 cos 

dx 2 ~ dxx 2 -\- y 2 "" (x 2 -f- y 2 ) 2 ~ r 2 

Substituting these values and those before found in the 
general expression above, we obtain 

d 2 u sin 2 6 du sin 6 cos 6 du 

dx 2 ~ r dr r 2 dd 

_ d 2 u sin 6 cos 6 d 2 u sin 2 6 d 2 u , % 

+ COsV^-2- — -_ + __. (I ) 



397. The transformation of the expression 



d 2 u d 2 u 
dx 2 ' dy 2 



from rectangular to polar coordinates is of importance in 
mathematical physics. 

Since the effect of putting \tz — 6 in place of 6 in equations 
(1), Art. 394, is to interchange x and y, the expression for 
d 2 u 
dy'' 

by interchanging sin 6 and cos 6 and reversing the sign of dd. 
Hence 



d 2 u cos 2 6 du sin cos du 

2 ^2 ~Ja~T~ 



2 may, in this case, be derived from equation (1), Art. 396, 



dy 2 ~~ r dr r 2 dd 



. d 2 w sin 6 cos d 2 w cos 2 # d 2 w 

dr 2 ' r drdd [ r 2 dd 2 v y 



§ XXXI.] TRANSFORMA TIONS OF {7 2 . 387 

Adding equations (1) and (2), 

d 2 u d 2 u d 2 u 1 du 1 d 2 u 

dx 2 ~^df = & 2 +7fc + 7 2 ~dF 2 ' • • • (3) 

398. The corresponding symbol in three dimensions, when 
u is also a function of the third rectangular coordinate 2, has 
been denoted by J7 2 ; thus, 

7 ~ dx 2 + dy 2 ^ dz 2 ' 



Then, by equation (3), 



d 2 u d 2 u 1 du 1 d 2 u 
d? ^dr 2 +rdr+?dd 1 



If ff If W X tfff A \M l/V 

V U = 1^ +^ + 7^ + Ziltf' • • • (4) 



Now z and r constitute rectangular coordinates of P in the 
plane POZ, where OZ is the axis of z. Hence, denoting OP 
by p and POZ by 0, ^ and <jS are polar coordinates of P in this 
plane, and 

z = p cos <£, r =|0 sin ^ (5) 

Therefore equation (3) gives 

d 2 u d 2 u d 2 u 1 du 1 d 2 u 

^ 2 + d? = di 2 + ~ P Tp + J 2 d$ 2 ' ' * \() 

Also by equation (5), Art. 394 (since r in equations (5) takes 
the place of y), 

du du . du cos (j> 

_=_s,ntf + ^ — (7) 



388 FUNCTIONS OF TWO OR MORE VARIABLE [Art. 398. 

Substituting from equations (6) and (7) and eliminating r, 
equation (4) becomes 

d 2 u 2 du 1 d 2 u cot (f> du 1 d 2 u 

V U = Jp 2 + ~pdp + J 2 d^ 2 + "T^^ 4 " p 2 sin 2 W 2 ' (8) 

Equation (4) gives y 2 u in the " cylindrical " coordinates 
r, 6, z; and equation (8), in the spherical coordinates p, d, <j>, 
where 

x = p sin cf> cos dy y = p sin <f) sin (9, z = p cos <£. 



Symbolic and Extended Forms of Taylor s Theorem. 

d 
399. A polynomial symbol involving D, where D= -=-> may 

be written in the form F(D) where F is a given rational in- 
tegral function. If F(D) is a function which can be developed 
in powers of D, it is regarded as equivalent to this series of 
symbols. Thus, by the exponential series, 

r • k 2 D 2 h B D 3 "1 



=Ax)+A(x)h+/"(x)- f +/'"(x)- ] + 



h 2 hs 



which is by Taylor's theorem the expansion of f(x-\-h) in 
powers of h. Hence Taylor's theorem can be expressed in 
the symbolic form 

f(x+h) = e hD f(x) (1) 

4-00. When the operand is a function of y also, this gives 
f(x+h, y) = e hD f(x, y); 



§ XXXI.] EXTENDED FORM OF TA YLOR'S THEOREM. 389 

d 
again, putting £)' = —, we have in like manner 

f(x+h, y+k) = e kD 'e hD f(x, y). ... (2) 
Now the equation 

is an algebraic identity for D and D' ', that is, it shows that the 
product of the two series represented by the first member is 
equivalent to the single series represented by the second 
member. Therefore, because D and jy are commutative 
symbols, equation (3) is, by Art. 387, a symbolic identity. 
Thus equation (2) becomes 

f(x+h, y+k) = e hT * +k ~*~y/(x, y), 
which,* when written out in full, takes the form 

f(x+h, y+k)=f(x, y ) + d £h+^k 



■h^w *&*+#»]+ 



The result is readily extended to any number of inde- 
pendent variables. 

Lagrange s Theorem. 

401. Let y be an implicit function of the independent vari- 
ables x and z, satisfying the relation 

y=z+x<p(y), (1) 



390 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 401. 

in which <p denotes any function; then, if we have 

»=/(y)» (2) 

u will also be a function of x and z. 

If now it be required to develop u in a series involving 
powers of x t we obtain by the application of Maclaurin's 
theorem 

d?ul x 2 

-J\+'-> • • • (3) 



U=u °+Tx 



X+ dx 



in which the coefficients are functions of z but not of x. We 

. - , , . . du d 2 u 

proceed to transform the derivatives—, -7-3, etc. into expres- 
sions in which z is the independent variable, before determining 
their values when x=o. 

Differentiating equation (1), we obtain the partial deriva- 
tives 

dy <P(x) dy 1 

dx~ i—x<p\y)' dz~i—x<p'(y)' 

hence, by equation (2), 

du = f(y)^(y) ^ du = f'{y) . 

dx i—x(p'(y) dz 1— x<P'(y)' ' ' W 



whence 



du , / du 

S=^5 (5) 



402. In order to deduce the required expressions for the 
higher derivatives, we first establish the following general 
theorem: 



§ XXXI.] LAGRANGE'S THEOREM. 391 

When y is a function of x and 2, and u and tp(y) are any 
functions of y, we have 

d .du d du 



To prove this theorem, we have only to perform the differ- 
entiations ; thus, putting f(y) for u } each member of (6) re- 
duces to 

du 
Substituting, in the general theorem (6), the value of -=- 

given in equation (5), we have 

d du d du 

d#®*=d;*®-*to* (7) 



d 
Applying the symbol — to equation (5), and reducing the 

second member by means of equation (7), we find 



d 2 u d r , . n0 du 

&=toto<y)?& • .... (8) 



Again, applying — to equation (8), we have 



d B u_ d d _ du__ d d \-\ 2 ^ u 



39 2 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 402. 



and, reducing by equation (7), 



d 3 u d 2 r , .„ du 



dx 3 dz- 



dz 



By successive repetitions of this process we obtain in general, 

d n u d n ~\^ / Nn du 

^[^Wl M r- ..... (9) 



dx n dz' 



dz 



403. In determining the values which these derivatives 
assume when #=o,we notice that, when # = 0, equation (1) gives 

y=z; hence u =f(z), and from equation (4), 



dz 



= f(z). 



Moreover, since the differentiations indicated in the second 
member of equation (9) have reference only to z, we may, in 
this equation, assign to x its value before the differentiations 
are effected: therefore 



du 
dx 



=mr<>). d £]rU [,p(zW ' {z) \' 



d n u ~ 
dx" 



|^[ (s)]V'(*H. 



Substituting these values in equation (3), we obtain 

Ay)=Az)+x4>{z)f{z) + ~ \[<P{z)ff {z) }+• • • 

rgn An — 1 C ) 



This result is known as Lagrange 's Theorem. 

404. As an application, we expand the function y deter- 
mined by 

y=z-\-xe y . 



§ XXXI.] DEVELOPMENTS BY LAGRANGE'S THEOREM. 393 

In this example <p(y) = e y , and f(y) = y, whence/ 7 ^) = I. The 
general term is therefore 



x n d n ~ 1 x n 

whence 



pfiz — mU — i s>nz . 

n\ dz n ~ l n\ ' 



y=z+e* . x +2e 2 *.^+3 2 e^+- . . 

To obtain the development of the function given by 

y= iJ r xe y J 

we put z=i in the preceding development. 

405. When the given relation between x and y is not in 
the form required for the application of Lagrange's theorem, 
an algebraic transformation sometimes enables us to make the 
application. Thus, if we have 

log y=xy (i) 

to develop y in powers of x, we put \ogy=y f ; whence equa- 
tion (i) becomes 

y=xe y (2) 

The latter equation being in the required form (but with 
z = o), we have 

u=zy=e y ', f'(y f )-= e y ' and <p(y) = e/. 
Hence the general term is 

y» d n ~ I x n 

e (n+*)z = / w+I \«-ig(«+i)s 

n\ dz n ~ I n\ } 



394 FUNCTIONS OF TWO OR MORE VARIABLES, [Art. 405. 

and putting z—O, we have 

/y»2 /y» 3 /y*4 

406. Lagrange's theorem may, in fact, be applied to f(y) 
whenever the relation between y and x is of the form 

y=F[z+xcp(y)]; 
for, if we put 

i =z -\-xcp{y), we have y=F(t); 

whence 

u=/F(f), and *=z+*0,F(/). 

Lagrange's theorem is therefore immediately applicable, 
the functions fF and cpF taking the place of /and <p in the 
development. Hence, substituting, we have 

m =m» +**m d -^+U z \i*pv^ f + • • • 

This form of the series is called Laplace's Theorem. 

The example in Art. 405 maybe regarded as a case of this 
theorem; for the given equation may be written in the form 

y— e xy y 

and we have in this case/(v) = v, also 

F(t) = e f , t=xy, z=o, and 0(y)—y. 

Both fF(z) and <pF(z) reduce to e z > and are identical with f(z) 
and 0(s) of Art. 405. 



XXXI.] examples. 395 



Examples XXXI. 

i. Change the independent variable from x to z in the equation 



d 2 y dy 

x 2 -— , 4- x— — \- y = o, when x = e z . 

ax* ax 



dy 

dz 



2 +^ = °' 



2. Change the independent variable from x toy in the equation 



dx 2 ^~ 2y \dx) °" 






3. Change the independent variable fromjy to x in the equation 

d 2 u du 

(i — v') -r-= — y - — h « = o, when v = sin a:. 

v ' dy 2 dy ' 

4. Change the independent variable from x to z in the equation 

, d 2 y dy a 2 , 1 

^ 2 — ■ + 2^r — - -j -y = o, when x = — . 

dx 2 dx x 2 ^ z 

5. Change the independent variable from x to z in the equation 

x * % + 3 * 2 z? + *ll +-' = °> when z = log *■ 

6. Change the independent variable from x to / in the equation 

d?y 1 dy 

&>+**c +J, = > when x=4f - 



39^ FUNCTIONS OF TWO OR MORE VARIABLES. [Ex. XXXI. 

d 2 u 
7. Given x = t -j- / 2 , transform — into an expression in which x 

is the independent variable. 

d 2 u d 2 u du 



dt 2 J dx 2 dx 

8. Change the independent variable from x to 2 in the equation 

d 2 y dy 

C 1 + * 2 ) -^ + * ^ + 2a * = °> when * = lo s D* + Vi 1 + * 2 )]. 

d 2 y 

_ +a(e ._ e - )==0 . 

9. Change the independent variable from ^ to 2 in the equation 

, o o ^^ a 2 dy x 2 , , 

(<z 2 — .r 2 ) —4; ;- H =0, when ^ 2 4. 2 2 = a 2 . 

dx 2 x dx a ' 

10. If ^ = e 6 , prove the symbolic identities 

d d _ r d d 

x Jx~ = Jv * l r dx x " = dd +r; 

and thence, more generally, 

*^£,*--«=[|+r}[^+r-i][|+r-,]... 

[J+ r -* +i >' 

in which r admits of negative and fractional values. 

11. Given x = e 9 , transform by the result of Ex. 10 the equation 

<py d 2 y dy 

X ^+S X dx^- 2X dx +2y =°- 

[i~ I ]ti +2 ]- >/=0 - 

12. When x = e 9 , prove by means of Ex. 10 

d* . v# 

x* 



& , r « n 



</* 2 L</0 2 - 4 

and verify when « = sin a*. 



§ XXXI.] EXAMPLES. 397 

13. When x = e 9 , show that 

14. Given .* = a (1 — cos /) and y = <2 (»/+ sin ')> prove that 

a 72 ^ » cos / -j- 1 

<&r 2 a sin 3 / 

15. Given ^ = r cos # andjy = r sin #, jy being a function of .*, 
prove that 



h+M^'-S 



dx 2 ~ I a dr . A3 

cose/ — — r sin # 



^ 2 z/ 
16. Find 'the general value of in terms of derivatives with 

respect to r and # (see Art. 395); also the special value, in the case of 
rectangular and polar coordinates. 

dhi du d 2 r du d?d d 2 udr dr 
dxdy dr dx dy dd dx dy dr 2 dx dy 

d?u rdr dd dr dd~\ <Pu dd dd ^ 
~T~ drddldx lp~ ~dj> dx\ ~^dd~ 2 dxdy'' 

(Pu sin 6 cos 6 du sin 2 6 — cos 2 6 du 



dxdy r dr r 2 dd 

, • n nd* u . cos 2 d— sin 2 d d 2 u sin d cos d d 2 u 
+ sm0cos0- 2 + — g _-. 

17. Given x = rcos d andy = r sin d, prove that 

du du du du du du 

x- y— =z — - and that x- — \- y — = r — . 

dy dx dd dx dy dr 



39 8 FUNCTIONS OF TWO OR MORE VARIABLES. [Ex. XXXL 



that 



18. If £, = x cos a — y sin a, and -q = x sin a -\-y cos a, prove 



(Pu <Pu <Pu 



cfiu 



dx 2 + dy 2 dB, 2 + df 

thence show that p 2 u is unchanged in value by any change in the 

rectangular planes of reference. 

d 2 u dPu 
ig. Transform the expression r 2 — „ 4- ttto into a function in which 

dr l da 4, 

x and ^ are the independent variables, having given x = r cos 6 and 

y = r sin 6. 



2 d 2 u d 2 u , „ „ 






</« 



du 



— x y — . 

dx dy 



20. If s = e? -j- **> an d ^ = £ _;c + £ _>, > prove that 



cPu (Pu d 2 u 

37.3 ~t~ 2 yznr. ~r 



</.* 



dx dy dy 



2 ~ S \s 2 ' ' 2S/ ds 



d 2 u 2 </ 2 « du du 



2i. If x = ^^ cos (£, and >* == #£# sin <£, prove that 



<^r 2 



d 2 u 



n d 2 u d 2 u du 



2Xy dx dy + * 2 dy* ~ d<j> 2 + dd 



22. Given v = z -j- .re^, expand e my in powers of #. 



x l 



e my _ ^ _|_ me (p + m)z x _|_ m ( 2 p _j_ ^(ap + «)* 1_ 



2! 



jr 



-fw(«/ + m) M - l e (H t + m)z — + 



23. Given jy = a -|- ^ 3 , expand j/ in powers of x. 

y = a -\-a*x + 6a 5 — 7 -1- 9 • Sa 7 — - -f- i2«n-ioa 9 — + 
2 1 3 • 4 • 



+ 



(3")' 

(2« -f- 1) ! #! 



.* M + 



§ XXXI.] EXAMPLES. 399 

y z — i 
24. Given v = z -f x , expand^ in powers of x. 



y + \(# - i)x + \z(# - i)a* + i( 5 2 4 - 6s 2 + i)*3 + 

25. Given v = a -\- by m , expand^ in powers of b. 

5* b 3 
y = a + a m b -f- 2tna 2m - x — + 3^(3^ — i)tf 3W_2 f- 

26. Given x 5 -j- 4^ -[- 2 = o, determine the value of x. 

i,i 5 , 35 



x 



, "r 7 13 ~t~ ,19 



27. Given_y = -J- jry 3 , expand^ 3 in powers of x. 

y 3 = a 3 + 3<z 5 x + 8<z 7 -^p + n.io^ 9 ^4-i4-i3-i2^ u ^-H 

28. Given j/ = £ + ^ log^, where e is the Napierian base, expand 
y in powers of x. 

A 

2e 2 6e l 



••V ••V- ••V 

j, = * + * + _+ 3 _ _ + 



29. Given _y = xe~ y , expand sin (a -\-y) in powers of x. 

Solution : 

The coefficient of — - is ^— ^ [>-** cos (a + z)~\; in which 2 is to 
n\ dz H * 

be put equal to zero after the differentiations. By the method of 

Art. 103, we find 

j n —i »— 1 

— —[e- MZ cos («+£)]=( — i) n-I (i-r-« 2 )~cos[a:+2— (» — i)cot _I «]; 



400 FUNCTIONS OF 7 'WO OR MORE m VARIABLES. [Ex. XXXI. 

hence 

sin(a -\-y) = sin a + x cos a + • • • 

— (— i) w (i + « 2 ) 2 cos [« — (»— i) cot -1 «] — • • • 

30. Develop [1 + Vi 1 ~~ e2 )~\~* m powers of e. 

e 2 
Put E = 1 -f- |/(i — e 2 ), whence E ' = 2 — =. 

r, + V(l _^i- P =± + /g. + Ai» + 3)«' , ^+4)(/+5K 



• • • 



INDEX. 



[The figures refer to the pages.] 



Acceleration, 88, 91 
Acnode, 273 ex. 10, 288 
Agnesi, 259 
Algebraic functions, 32 

curves, 249 et seq., 263 et seq. 

equations of epicyclics, 284 
Analytical triangle, 253 foot-note 
Archimedes, spiral of, 309, 312 ex. 9 
Argument of complex quantity, 221 
Astroid, see four-cusped hypocycloid, 286 
Asymptote, 262, 270, 304 

equation of, 264 
Asymptotic circle, 307 

parabola, 270 foot-note 
Attwood's machine, 16 
Auxiliary curves, 250 

variable, 274, 286 

Bernoulli, lemniscate of, 296 
Bernoulli's numbers, 237 
Binomial Theorem, 182 
Branches, infinite, 267 
parabolic, 269 

Cardioid, 285, 302, 311 ex. 6, 344 ex. 

Cassinian ovals, 313 exs. 29, 30 
Catenary, 217 foot-note, 261 ex. 9, 332, 
338 ex. 6, 341 ex. 24, 342 ex. 32' 

evolute and involute of, 334 
Caustics, 355, 360 ex. 27 
Centre of curvature, 316 

locus of, 319 
Change of independent variable, 380 et 

seq 
Circular functions, 61 
Cissoid of Diodes, 261 ex. 18, 338 ex. 8, 

357 ex. 6 
Commutative operations, 372 



Complex quantities, 221 

functions of, 222 
Complex unit, 222 

Component velocity and acceleration, 91 
Computation of e, 181 

of Napierian logarithms, 196 

of it, 207 
Conchoid of Nicomedes, 261 ex. 20, 313 

ex. 16 
Conjugate complex quantities, 222 
Constant rates, 15 
Continuous variables, I 

functions, 5 
Continued products, 232 et seq., 242 
Contrary flexure, points of, 97 
Convergent series, 1 73 et seq. 
Coordinates in terms of a third variable, 

274 
Courbe du Diable, 290 ex. 14 
Critical points, 133, 135, 294 
Cubical parabola, 133, 252, 339 ex. 12 
Curtate cycloid, 278 
Curvature, 97, 314 (see also Radius of) 

centre of, 316 

circle of, 315 

measure of, 314 
Curve tracing, 255, 266, 287, 295 
Cusps, 133, 276, 320 

ramphoid, 273 ex. 19, 340 ex. 19 
Cusp locus, 349 foot-note 
Cycloid, 275, 316, 327 

DeMoivre's Theorem, 223 
Derivatives, 26 

graphic representation of, 27 

sign of, 28 

of implicit functions, 82, 98, 165 

second, 89, 96 

successive, 95, 367 



402 



INDEX. 



Derivatives, wth, expressions for, 103 

infinite values of, 132 
Descartes, folium of, 273 ex. 11 
Development in series, 172 . 

of <p cot <p, 240 
Differences, finite, 29 foot-note, 363, 371 

limiting ratio of, 29 
Differentials, 18 
Differential of a sum, 20 

a multiple, 21 

the square, 32 

the product, 22, 35 

the reciprocal, 41 

the quotient, 42 

the logarithmic function, 49 

the exponential function, 56 

the trigonometric functions, 62 et seq. 

the inverse trigonometric functions, 70 
et seq. 

a function of two variables, 80, 362 
Differential equations, 202, 210 

triangle, 28, 62 
Differentiation, 32 

of an inverse function, 35 

of a function of a function, 36 

of implicit functions, 82, 98, 165 

logarithmic, 54 

successive, 95 
Diocles, cissoid of, 261 ex, 18 
Discontinuity of functions, 6, 156 
Discriminant, 348, 351 
Discrimination of maxima and minima, 

122 
Distributive operations, 374 
Divergent series, 173 
Double generation of epitrochoids and 

hypotrochoids, 282 
Double points, 166, 168, 272 
Dygogram, 301 

Ellipse, 248, 274, 285, 311 ex. 5, 319, 

320, 322, 342 ex. 28 
Envelopes, 346 
Epicyclics, 281 

Epicycloid, 278, 290 ex. 8, 342 ex. 34 
Epitrochoid, 278, 289 ex. 4 
Equiangular spiral, 310 
Euler's series, 206 

theorems in homogeneous functions, 376 
Evaluation of indeterminate forms, 143, 

198 
Even functions, 214 



Evolute, 319, 324, 353 

of the ellipse, 320 

of the cycloid, 327 

of the catenary, 334 
Explicit functions, 4 
Exponential curve, 57 

functions, differentiation of, 56 

Focus of a caustic, 355 

Folium, 273 ex. 11 

Formula for non-linear functions of t, 19 

Formulae of differentiation, 32, 78 

Four-cusped hypocycloid, 286, 290 exs. 

6, 7, 337 ex. 2, 352 
Functional equations, 10 
Functions, 2 

implicit, 3, 82, 98, 129 

increasing and decreasing, 8 

inverse, 4 

linear, 9 

periodic, 61 

rational integral, 172 
Functions which vanish with x y 147 

of two variables, 3, 80, 134, 163, 369 

of pure imaginaries, 218 

of complex variables, 222 

General term of development, 201 
Generating function, 173 
Geometric variables, rates of, 37 

maxima and minima of, 114 
Gradient, 29 
Graph of a function, 4 
Graphic representation of the derivative, 

27, 365 
Gregory's series, 207 

Homogeneous (differential) formulae, 75 
Homogeneous functions, 376 
Huyghens, 328 
Hyperbola, 311 ex. 4, 312 ex. 10, 339 exs. 

11, 14, 341 exs. 27, 28 
Hyperbolic functions, 216 

Illusory, see Indeterminate forms 
Imaginary quantities, 218 

roots of unity, 226 
Implicit functions, 3 

derivatives of, 82, 98, 165 

higher derivatives of, 98 

maxima and minima of, 129 



INDEX. 



403 



Increments, 15, 17 

limiting value of ratio of, 18, 29 
Independent variable, 2, 25 

change of, 380 
Indeterminate forms, 141 ; 0/0, 143 ; 00 /oo , 
I5 1 * 153; °Xoo, 155; 00-00,158; 
i°°, 161; oo°, 162; o°, 163 

of derivatives, 165 

of functions of two variables, 163 
Infinite branches of a curve, 267 

values of r, 303 

values of the derivative, 132 
Infinitesimals, 29 foot-note, 148 
Infinity, points at, 260, 303 
Inflexion, points of, 97, 122, 133, 261 ex. 

19, 296, 305, 312 ex. II 
Instantaneous centre, 320 
Intrinsic equations, 331 

of e volutes, 333 
Inverse functions, 4 

trigonometric functions, 70 
Involutes, 334 

of the circle, 337, 360 ex. 25 
Isochronism, 328 

Lagrange's expression for the, remainder, 
192 

Theorem, 389 
Laplace's Theorem, 394 
Leibnitz' Theorem, 106 
Lemniscate of Bernoulli, 296, 311 ex. 

I, 313 ex. 29, 341 ex. 27, 353 
Limaeon of Pascal, 298, 341 ex. 26 
Limiting ratio of differences, 18, 29, 369 

values of discontinuous functions, 156 

values of vanishing fractions, 141 
Limits of convergence, 175 
Linear functions, 9, 16 
Lituus, 311 ex. 7, 342 ex. 29 
Locus of centre of curvature, 319 
Logarithmic curve, 53, 339 ex. 10 

differentiation, 54 

function, differentiation of, 49 

trigonometric functions, 66 

spiral, 310 
Loxodromic curve, 310 foot-note 

Maclaurin's Theorem, 179 
Maxima and minima, 112, 197 

abscissae, 132 

alternation of, 124 

coordinates, 268 



Maxima and minima, discrimination of, 
122 

of functions of two variables, 134 

of implicit functions, 129 

of substituted functions, 126 
Modulus of complex quantity, 221 
Multiple-valued functions, 5, 223, 225 

Napierian logarithms, 52 

base, 52, 162, 181 
Negative pedals, 356, 360 
Nicomedes, conchoid of, 261 ex, 20, 313 

ex. 16 
Node-locus, 349 foot-note 
Non-commutative operations, 373 
Non-linear function, 19 

differential equations, 210 
Normal, 247, 249, 254 

envelope of, 353 

One-valued functions, 5, 227 foot-note 
Operand, 108, 367 
Ovals, 336 

Cassinian, 313 exs. 29, 30 

Parabola, 4, 31 ex. 8, 34 foot-note, 337 
et seq. exs. I, 5, 9, 31 

of nth. degree, 25 1 

cubical, 133, 252 ' 

semi-cubical, 134, 252 
Parabolic branches, 269 
Parallel curves, 335 
Parameters, variable, 344, 352 
Partial differential, 80, 362 

derivative, 80, 364 
Pascal, limaeon of, 298 
Pedal, 302, 330 
Pencils of curves, 345 
Perpendicular on the tangent, 255, 293, 

330 

Points at infinity, 260, 303 

Points of inflexion, 97, 122, 133, 296, 

305, 312 «f.-ii 
Polar coordinates, 291 et seq. 

subtangent and subnormal, 293 
Primary values of inverse trigonometric 
functions, 70 

of the argument, 221 

of the logarithm, 223 
Prolate cycloid, 278, 289 ex. 3, 339 ex. 13 
Protractions, 298, 309, 313 ex. 16 



404 



INDEX. 



Quadratic factors, 227, 231 ex. 27 

Radius of curvature, 315 et seq. 
at critical points, 317 
in rectangular coordinates, 321 
in polar coordinates, 328 
of the cardioid, 344 ex. 41 
catenary, 338 ex. 6 
cissoid, 338 ex. 8 
cubical parabola, 339 ex. 12 
cycloid, 317 

ellipse, 320, 322, 342 ex. 28 
epicycloid, 342 ex. 34 
four-cusped hypocycloid, 337 ex. 2 
hyperbola, 339 exs. II, 14, 341 ex. 27 
lemniscate, 341 ex. 27 
logarithmic curve, 339 ex. 10 
limaeon, 341 ex. 26 
lituus, 342 ex. 29 
parabola, 337 ex. 1, 338 ex. 5, 339 ex. 

9, 342 ex. 31 
prolate cycloid, 339 ex. 13 
strophoid, 340 ex. 18 
Radical axis, 346 

Ramphoid cusp, 273 ex, 19, 340 ex. 19 
Rates of variation, 14 
constant, 15 
not uniform, 17 

of a function of an independent vari- 
able, 25 
of geometrical magnitudes, 37 
Ratio of differences, 18, 29, 369 
Rational integral function, 172 
Recapitulation of differential formulae, 78 
Reciprocal, differential of, 41 

of radius vector, 292, 306 
Reciprocal spiral, 309 
Rectifiable curves, 332 
Relative rates, 25 

Remainder in infinite series, 174, 190 
Roberval, 289 
Roots of unity, 226 

Semi-cubical parabola, 134, 252 
Series in ascending powers of x, 172 

convergent and divergent, 173 

differentiation of, 177 

Euler's, 206 

exponential, 180 

Gregory's, 207 



Series, hyperbolic, 216, 243 

inverse hyperbolic, 230 

inverse trigonometric, 185 ex. 4, 212 

logarithmic, 194, 195 

trigonometric, 183, 184, 240, 245 
Slope, 8, 29, 34 foot-note 
Spirals, 307 

of Archimedes, 309 

reciprocal, 309 

logarithmic or equiangular, 310 
Stop- point, 157, 162 foot-note 
Strophoid, 257, 305, 340 ex. 18, 356 
Subtangent and subnormal, 253, 293 
Sum of infinite series, 174 
Symbolic identities, 375 
Symbols of operation, 27, 96, 372 
Systems of curves, 344 

Tangent, 28, 247, 249, 254 
Taylor's Theorem, 187, 388 

remainder in, 190 
Total differential, 80, 362 

derivative, 365 foot-note, 366 
Tracing of curves, 255, 266, 287, 295 
Tractrix, 336 

Transformation to new independent vari- 
ables, 382 et seq. 

symbolic, 375, 396 
Trigonometric functions, 61 
Trochoid, 278 
Two-cusped epicycloid, 289 ex. 5, 355, 

359**. 18 
Two- valued functions, 5, 230**. 15 

Ultimate intersections, 347 
Uneven functions, 214 
Units, complex, 222 
Unity, roots of, 226 

Vanishing fractions, 142 
Variable velocity, 16 

rate, 17 
Variables, I, 17 
Velocity, 16, 88, 91 

Wallis' expression for Tt, 235 
Witch, 259 

Zero values of r, 295 



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Gore's Elements of Geodesy 8vo, 2 50 

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Hering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 50 

Howe's Retaining Walls for Earth nmo, 1 25 

Johnson's Theory and Practice of Surveying Small 8vo, 4 00 

Statics by Algebraic and Graphic Methods 8vo, 2 00 

Kiersted's Sewage Disposal nmo, 1 25 

Laplace's Philosophical Essay on Probabilities. (Truscott and Emory.) nmo, 2 00 

Mahan's Treatise on Civil Engineering. (1873 ) (Wood.) 8vo, 5 00 

* Descriptive Geometry 8vo, x 50 

Merriman's Elements of Precise Surveying and Geodesy 8vo, 2 50 

Elements of Sanitary Engineering 8vo, 2 00 

Merriman and Brooks's Handbook for Surveyors. . .* i6mo, morocco, 2 00 

Nugent's Plane Surveying , . . 8vo, 3 50 

Ogden's Sewer Design nmo, 2 00 

Patton's Treatise on Civil Engineering 8vo half leather, 7 50 

Reed's Topographical Drawing and Sketching 4to, 5 00 

Rideal's Sewage and the Bacterial Purification of Sewage 8vo, 3 50 

Siebert and Biggin's Modern Stone-cutting and Masonry 8vo, z 50 

Smith's Manual of Topographical Drawing. (McMillan.) 8vo, 2 50 

Sondericker's Graphic Statics, wun Applications to Trusses, Beams, and 

Arches 8vo, 2 00 

Taylor and Thompson's Treatise on Concrete, Plain and Reinforced. (In press.) 

* Trautwine's Civil Engineer's Pocket-book i6mo, morocco, 5 00 

Wait's Engineering and Architectural Jurisprudence 8vo, 6 00 

Sheep, 6 50 

Law of Operations Preliminary to Construction in Engineering and Archi- 
tecture 8vo, s 00 

Sheep, 5 50 

Law of Contracts 8vo, 3 00 

Warren's Stereotomy — Problems in Stone-cutting 8vo, 2 50 

Webb's Problems in the U«e and Adjustment of Engineering Instruments. 

1 6mo, morocco, 1 25 

* Wheeler's Elementary Course of Civil Engineering 8vo, 4 00 

Wilson's Topographic Surveying 8vo, 3 50 

BRIDGES AND ROOFS. 

Boiler's Practical Treatise on the Construction of Iron Highway Bridges. .8vo, 2 00 

* Thames River Bridge 4to, paper, 5 00 

Burr's Course on the Stresses in Bridges and Roof Trusses, Arched Ribs, and 

Suspension Bridges 8vo, 3 50 

Du Bois's Mechanics of Engineering. Vol. II Small 4to, 10 00 

Foster's Treatise on Wooden Trestle Bridges 4to, 5 00 

Fowler's Coffer-dam Process for Piers 8vo, 2 50 

Greene's Roof Trusses 8vo, 1 25 

Bridge Trusses 8vo, 2 50 

Arches in Wood, Iron, and Stone 8vo, 2 50 

Howe's Treatise on Arches 8vo, 4 00 

Design of Simple Roof-trusses in Wood and Steel 8vo, 2 00 

Johnson, Bryan, and Turneaure's Theory and Practice in the Designing of 

Modern Framed Structures. Small 4to, 10 00 

Merriman and Jacoby's Text-book on Roofs and Bridges: 

Part 1. — Stresses in Simple Trusses 8vo, 2 50 

Part II. — Graphic Statics 8vo, 2 50 

Part III. — Bridge Design. 4th Edition, Rewritten 8vo, 2 50 

Part IV. — Higher Structures 8vo, 2 50 

Morison's Memphis Bridge 4to, 10 00 

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Waddell's De Pontibus, a Pocket-book for Bridge Engineers. . . 1 6mo, morocco, 
Specifications for Steel Bridges i2mo, 

Wood's Treatise on the Theory of the Construction of Bridges and Roofs.8vo, 

Wright's Designing of Draw-spans: 

Part L — Plate-girder Draws 8vo, 

Part II. — Riveted-truss and Pin-connected Long-span Draws 8vo, 

Two parts in one volume 8vo, 

HYDRAULICS. 

Bazin's Experiments upon the Contraction of the Liquid Vein Issuing from an 

Orifice. (Trautwine.) 8vo, 

Bovey's Treatise on Hydraulics 8vo , 

Church's Mechanics of Engineering 8vo, 

Diagrams of Mean Velocity of Water in Open Channels paper, 

Coffin's Graphical Solution of Hydraulic Problems i6mo, morocco, 

Flather's Dynamometers, and the Measurement of Power 12 mo, 

Polwell's Water-supply Engineering 8vo, 

Prizell's Water-power 8vo, 

Fuertes's Water and Public Health i2mo, 

Water-filtration Works i2mo, 

Ganguillet and Kutter's General Formula for the Uniform Flow of Water in 

Rivers and Other Channels. (Hering and Trautwine.) 8vo, 

Hazen's Filtration of Public Water-supply 8vo, 

Hazlehurst's Towers and Tanks for Water- works 8vo , 

Herschel's 115 Experiments on the Carrying Capacity of Large, Riveted, Metal 

Conduits 8vo, 2 00 

Mason's Water-supply. (Considered Principally from a Sanitary Stand- 
point.) 3d Edition, Rewritten 8vo, 

Merriman's Treatise on Hydraulics, oth Edition, Rewritten 8vo, 

* Michie's Elements of Analytical Mechanics 8vo, 

Schuyler's Reservoirs for Irrigation, Water-power, and Domestic Water- 
supply Large 8vo, 

•* Thomas and Watt's Improvement of Riyers. (Post., 44 c. additional), 4to, 

Turneaure and Russell's Public Water-supplies 8vo, 

Wegmann's Desien and Construction of Dams 4to, 

Water-supply of the City of New York from 1658 to~i8o5 4to, 10 

Weisbach's Hydraulics and Hydraulic Motors. (Du Bois.) 8vo, 

Wilson's Manual of Irrigation Engineering Small 8vo, 

Wolff's Windmill as a Prime Mover 8vo, 

Wood's Turbines 8vo, 

Elements of Analytical Mechanics 8vo, 

MATERIALS OF ENGINEERING. 

Baker's Treatise on Masonry Construction 8vo, 

Roads and Pavements 8vo, 

Black's United States Public Works oblong 4to, 

Bovey's Strength of Materials and Theory of Structures 8vo, 

Burr's Elasticity and Resistance of the Materials of Engineering. 6th Edi- 
tion, Rewritten 8vo, 

Byrne's Highway Construction 8vo, 

Inspection of the Materials and Workmanship Employed Jn Construction. 

i6mo, 

Church's Mechanics of Engineering 8vo, 

Du Bois's Mechanics of Engineering. Vol. I Small 4to, 

Johnson's Materials of Construction Large 8vo, 

Keep's Cast Iron 8vo, 

Lanza's Applied Mechanics -. 8vo, 

Martens's Handbook on Testing Materials. (Henning.) 2 vols 8vo, 

Merrill's Stones for Building and Decoration 8vo, 

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Merriman's Text-book on the Mechanics of Materials 8vo, 4 00 

Strength of Materials nmo, 1 00 

Metcalf's Steel. A Manual for Steel-users nmo, 2 00 

Patton's Practical Treatise on Foundations 8vo, 5 00 

Richey's Hanbbook for Building Superintendents of Construction. (In press.) 

Rockwell's Roads and Pavements in France nmo, 1 25 

Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo, 3 00 

Smith's Materials of Machines nmo, 1 00 

Snow's Principal Species of Wood 8vo, 3 50 

Spalding's Hydraulic Cement i2mo, 2 00 

Text-book on Roads and Pavements nmo, 2 00 

Taylor and Thompson's Treatise on Concrete, Plain and Reinforced. (In 
press.) 

Thurston's Materials of Engineering. 3 Parts 8vo, 8 00 

Part L — Non-metallic Materials of Engineering and Metallurgy 8vo, 2 00 

Part II. — Iron and Steel 8vo, 3 50 

Part III. — A Treatise on Brasses, Bronzes, and Other Alloys and their 

Constituents 8vo, 2 50 

Thurston's Text-book of the Materials of Construction 8vo, 5 00 

Tillson's Street Pavements and Paving Materials 8vo, 4 00 

Waddell's De Pontibus. (A Pocket-book for Bridge Engineers.) . . i6mo, mor, , 3 00 

Specifications for Steel Bridges nmo, x 25 

Wood's Treatise on the Resistance of Materials, and an Appendix on the Pres- 
ervation of Timber 8vo., 2 00 

Elements of Analytical Mechanics 8vo, 3 00 

Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. . .8vo, 4 00 

RAILWAY ENGINEERING. 

Andrews's Handbook for Street Railway Engineers. 3X5 inches, morocco, z 25 

Berg's Buildings and Structures of American Railroads 4 to, 5 00 

Brooks's Handbook of Street Railroad Location i6mo. morocco, 1 50 

Butts's Civil Engineer's Field-book i6mo, morocco, 2 50 

Crandall's Transition Curve i6mo. morocco, 1 50 

Railway and Other Earthwork Tables. 8vo, 1 50 

Dawson's "Engineering" and Electric Traction Pocket-book. i6mo, morocco, 5 00 

Dredge's History of the Pennsylvania Railroad: (1879) Paper, 5 00 

* Drinker's Tunneling, Explosive Compounds, and Rock Drills, 4to, half mor., 25 00 

Fisher's Table of Cubic Yards Cardboard, 25 

Godwin's Railroad Engineers' Field-book and Explorers' Guide i6mo, mor., 2 50 

Howard's Transition Curve Field-book i6mo, morocco. 1 50 

Hudson's Tables for Calculating the Cubic Contents of Excavations and Em- 
bankments 8vo, 1 00 

Molitor and Beard's Manual for Resident Engineers i6mo, 1 00 

Nagle's Field Manual for Railroad Engineers i6mo, morocco. 3 00 

Philbrick's Field Manual for Engineers i6mo, morocco, 3 00 

Searles's Field Engineering i6mo, morocco, 3 00 

Railroad Spiral i6mo, morocco, z 50 

Taylor's Prismoidal Formulae and Earthwork 8vo, 1 50 

* Trautwine's Method of Calculating the Cubic Contents of Excavations and 

Embankments by the Aid of Diagrams 8vo, 2 00 

The Field Practice of [Laying Out Circular Curves for Railroads. 

1 2mo, morocco, 2 50 

Cross-section Sheet Paper, 25 

Webb's Railroad Construction. 2d Edition, Rewritten i6mo. morocco, 5 00 

Wellington's Economic Theory of the Location of Railways Small 8vo, 5 00 

DRAWING. 

Barr's Kinematics of Machinery 8vo, 2 50 

* Bartlett's Mechanical Drawing 8vo, 3 00 

* " Abridged Ed 8vo, 1 5» 

8 



Coolidge's Manual of Drawing 8vo, paper, i oo 

Coolidge and Freeman's Elements of General Drafting for Mechanical Engi- 
neers. (In press.) 

•rley's Kinematics of Machines 8vo, 

i's Text-book on Shades and Shadows, and Perspective 8vo, 

.oiison's Elements of Mechanical Drawing. (In press.) 
ones's Machine Design: 

Part I. — Kinematics of Machinery 8vo, 

Part II. — Form, Strength, and Proportions of Parts 8vo t 

MacCord's Elements of Descriptive Geometry . , 8vo, 

Kinematics; or, Practical Mechanism , 8vo, 

Mechanical Drawing , 4to, 

Velocity Diagrams 8vo, 

• Mahan's Descriptive Geometry and Stone-cutting 8vo, 

Industrial Drawing. (Thompson.) 8vo> 

Mover's Descriptive Geometry. (In press.) 

Reed's Topographical Drawing and Sketching 4to, 

Reid's Course in Mechanical Drawing 8vo, 

Text-book of Mechanical Drawing and Elementary Machine Design. .8vo, 

Robinson's Principles of Mechanism 8vo, 

Smith's Manual of Topographical Drawing. (McMillan.) 8vo, 

Warren's Elements of Plane and Solid Free-hand Geometrical Drawing. . 12 mo, 

Drafting Instruments and Operations iamo, 

Manual of Elementary Projection Drawing iamo, 

Manual of Elementary Problems in the Linear Perspective of Form and 

Shadow nmo, 

Plane Problems in Elementary Geometry umo, 

Primary Geometry iamo, 

Elements of Descriptive Geometry, Shadows, and Perspective 8vo, 

General Problems of Shades and Shadows 8vo, 

Elements of Machine Construction and Drawing 8vo, 

Problems. Theorems, and Examples in Descriptive Geometry 8vo, 

Weisbach's Kinematics and the Power of Transmission. (Hermann and 

Klein.) 8vo, 

Whelpley's Practical Instruction in the Art of Letter Engraving iamo, 

Wilson's Topographic Surveying 8vo, 

Free-hand Perspective 8vo, 

Free-hand Lettering 8vo, 

Woolf 's Elementary Course in Descriptive Geometry Large 8vo, 

ELECTRICITY AND PHYSICS. 

Anthony and Brackett's Text-book of Physics. (Magie.). ...... .Small 8vo, 

Anthony's Lecture-notes on the Theory of Electrical Measurements iamo, 

Benjamin's History of Electricity 8vo, 

Voltaic Cell. 8vo, 

Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.). , 8vo, 

Crehore and Squier's Polarizing Photo-chronograph 8vo, 

Dawson's "Engineering" and Electric Traction Pocket-book. . x6mo, morocco, 
Dolezalek's Theory of the Lead Accumulator (Storage Battery). (Von 

Ende.) i2mo, ~ 

Duhem's Thermodynamics and Chemistry. (Burgess.) 8vo, 

Flather's Dynamometers, and the Measurement of Power nmo, 

Gilbert's De Magnete. (Mottelay.) 8vo, 

Hanchett's Alternating Currents Explained i2mo, 

Hering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 

Holman's Precision of Measurements 8vo, 

Telescopic Mirror-scale Method, Adjustments, and Testa. Large Svo, 

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Landauer's Spectrum Analysis. (Tingle.) 8vo, 

Le Chatelier's High-temperature Measurements. (Boudouard — Burgess. )i2mo, 
Lob's Electrolysis and Electrosynthesis of Organic Compounds. (Lorenz.) nmo, 

* Lyons's Treatise on Electromagnetic Phenomena. Vols. I. and II. »vo, each, 

* Michie. Elements of Wave Motion Relating to Sound and Light. . . 8vo, 

Niaudet's Elementary Treatise on Electric Batteries. (FishDack. ) nmo, 

* Rosenberg's Electrical Engineering. (Haldane Gee — Kinzbrunner.) . . . .8vo, 

Ryan, Norris, and Hoxie's Electrical Machinery. VoL L 8vo, 

Thurston's Stationary Steam-engines 8vo, 

* Tillman's Elementary Lessons in Heat 8vo, 

Tory and Pitcher's Manual of Laboratory Physics Small 8vo, 

Ulke's Modern Electrolytic Copper Refining 8vo, 

LAW. 

* Davis's Elements of Law 8vo, 

* Treatise on the Military Law of United States 8vo, 

* Sheep, 

Manual for Courts-martial i6mo, morocco, 

Wait's Engineering and Architectural Jurisprudence 8vo, 

Sheep, 

Law of Operations Preliminary to Construction in Engineering and Archi- 
tecture 8vo, 

Sheep, 

Law of Contracts 8vo, 

Winthrop's Abridgment of Military Law i2mo, 

MANUFACTURES. 

Bernadou's Smokeless Powder — Nitro-cellulose and Theory of the Cellulose 

Molecule i2mo, 2 50 

Bolland's Iron Founder i2mo, 2 50 

M The Iron Founder," Supplement 121110, 2 50 

Encyclopedia of Founding and Dictionary of Foundry Terms Used in the 

Practice of Moulding i2mo, 3 00 

Blssler's Modern High Explosives 8vo, 4 00 

Bffront's Enzymes and their Applications. (Prescott.) 8vo, 3 00 

Fitzgerald's Boston Machinist i8mo, x 00 

Ford's Boiler Making for Boiler Makers i8mo, x 00 

Hopkins's Oil-chemists' Handbook 3vo, 3 00 

Keep's Cast Iron 8vo, a 50 

Leach's The Inspection and Analysis of Food with Special Reference to State 

Control. (In preparation.) 
Matthews's The Textile Fibres. (In press.) 

Metcalf's SteeL A Manual for Steel-users x2mo, a 00 

Metcalfe's Cost of Manufactures — And the Administration of Workshops, 

Public and Private 8vo, 5 00 

Meyer's Modern Locomotive Construction .4to, 10 00 

Morse's Calculations used in Cane-sugar Factories x6mo, morocco, x 50 

* Reisig's Guide to Piece-dyeing 8vo, 25 00 

Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo, 3 00 

Smith's Press-working of Metals 8vo, 3 00 

Spalding's Hydraulic Cement 121110, 2 00 

Spencer's Handbook for Chemists of Beet-sugar Houses i6mo, morocco, 3 00 

Handbook tor sugar Manufacturers and their Chemists. . . x6mo. morocco, 2 00 

Taylor and Thompson's Treatise on Concrete, Plain and Reinforced. (In 
press.) 

Thurston's Manual of Steam-boilers, their Designs, Construction and Opera- 
tion 8vo, 5 00 

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* Walke's Lectures on Explosives 8vo, 

West's American Foundry Practice i2mo, 

Moulder's Text-book nmo, 

Wiedemann's Sugar Analysis Small 8vo, 

Wolff's Windmill as a Prime Mover 8vo, 

Woodbury's Fire Protection of Mills 8vo, 

Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. . .8vo, 

MATHEMATICS. 

Baker's Elliptic Functions 8vo, 

* Bass's Elements of Differential Calculus nmo, 

Briggs's Elements of Plane Analytic Geometry i2mo, 

Compton's Manual of Logarithmic Computations 12 mo, 

Davis's Introduction to the Logic of Algebra 8vo, 

* Dickson's College Algebra Large nmo; 

* Answers to Dickson's College Algebra 8vo, paper, 

* Introduction to the Theory of Algebraic Equations Large 12 mo, 

Halsted's Elements of Geometry 8vo, 

Elementary Synthetic Geometry 8vo, 

Rational Geometry i2mo, 

* Johnson's Three-place Logarithmic Tables: Vest-pocket size paper, 

100 copies for 

* Mounted on heavy cardboard, 8X10 inches, 

10 copies for 

Elementary Treatise on the Integral Calculus Small 8vo, 

Curve Tracing in Cartesian Co-ordinates nmo, 

Treatise on Ordinary and Partial Differential Equations Small 8vo, 

Theory of Errors and the Method of Least Squares i2mo, 

* Theoretical Mechanics i2mo, 

Laplace's Philosophical Essay on Probabilities. (Truscott and Emory.) i2mo, 

* Ludlow and Bass. Elements of Trigonometry and Logarithmic and Other 

Tables 8vo, 

Trigonometry and Tables published separately Each, 

* Ludlow's Logarithmic and Trigonometric Tables 8vo, 

Maurer's Technical Mechanics 8vo, 

Merriman and Woodward's Higher Mathematics 8vo, 

Merriman's Method of Least Squares 8vo, 

Rice and Johnson's Elementary Treatise on the Differential Calculus. Sm., 8vo, 

Differential and Integral Calculus. 2 vols, in one Small 8vo, 

Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo, 

Wood's Elements of Co-ordinate Geometry 8vo, 

Trigonometry: Analytical, Plane, and Spherical nmo, 

MECHANICAL ENGINEERING. 

MATERIALS OF ENGINEERING, STEAM-ENGINES AND BOILERS. 

Bacon's Forge Practice nmo, z 50 

Baldwin's Steam Heating for Buildings nmo, 2 50 

Barr's Kinematics of Machinery 8vo, 2 50 

* Bartlett's Mechanical Drawing 8vo, 3 00 

* " " " Abridged Ed 8vo, x s« 

Benjamin's Wrinkles and Recipes nmo, 2 00 

Carpenter's Experimental Engineering 8vo, 6 00 

! -Heating and Ventilating Buildings : 8vo, 4 00 

Gary's Smoke Suppression in Plants using Bituminous Coal. {In prep- 
aration.) 

Clerk's Gas and Oil Engine Small 8vo, 4 00 

Coolidge's Manual of Drawing 8vo, paper, 1 00 

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Coolidge and Freeman's Elements of General Drafting for Mechanical En- 
gineers. (In preaa.) 

Cromwell's Treatise on Toothed Gearing um. t 50 

Treatise on Belts and Pulleys 121.. 50 

Durley's Kinematics of Machines $\ \ 00 

Flather's Dynamometers and the Measurement of Power ,12011 j 00 

Rope Driving . 1 2 id. 2 00 

Gill's Gas and Fuel Analysis for Engineers = . . <^ ... 1 25 

Hall's Car Lubrication r 2 aio, 1 00 

Hering's Ready Reference Tables (Conversion Factors) i6mo, morocco, a 50 

Hutton's The Gas Engine 8vo, 5 00 

Jones's Machine Design: 

Part I. — Kinematics of Machinery 8vo, x 50 

Part II. — Form, Strength, and Proportions of Parts 8vo, 3 00 

Kent's Mechanical Engineer's Pocket-book i6mo, morocco, 5 00 

Kerr's Power and Power Transmission 8vo, a 00 

Leonard's Machine Shops. Tools, and Methods. (In preaa.) 

MacCord's Kinematics; or, Practical Mechanism 8vo, 5 00 

Mechanical Drawing 4to, 4 00 

Velocity Diagrams 8vo, 1 50 

Mahan's Industrial Drawing. (Thompson.) 8vo, 3 So 

Poole's Calorific Power of Fuels 8vo, 3 00 

Reid's Course in Mechanical Drawing 8vo. a 00 

Text-book of Mechanical Drawing and Elementary Machine Design. .8vo, 3 00 

Richards's Compressed Air iamo, 1 50 

Robinson's Principles of Mechanism 8vo, 3 00 

Schwamb and Merrill's Elements of Mechanism. (In press.) 

Smith's Press-working of Metals 8vo, 3 00 

Thurston's Treatise on Friction and Lost Work in Machinery and Mill 

Work 8vo, 3 00 

Animal as a Machine and Prime Motor, and the Laws of Energetics . iamo, 1 00 

Warren's Elements of Machine Construction and Drawing 8vo, 7 50 

Weisbach's Kinematics and the Power of Transmission. Herrmann — 

Klein. ) 8vo, 5 00 

Machinery of Transmission and Governors. (Herrmann — Klein.). .8vo, 5 00 

HydrauLcs and Hydraulic Motors. (Du Bois.) 8vo, 5 00 

Wolff's Windmiil as a Prime Mover 8vo, 3 00 

Wood's Turbines 8vo, a 50 

MATERIALS OF ENGINEERING. 

Bovey's Strength of Materials and Theory of Structures 8vo, 7 30 

Burr's Elasticity and Resistance of the Materials of Engineering. 6th Edition, 

Reset. 8vo, 7 50 

Church's Mechanics of Engineering 8vo, 6 00 

Johnson'" Materials of Construction Large 8vo, 6 00 

Keep's Cast Iron 8vo, a 50 

Lanza's Applied Mechanics 8vo, 7 50 

Martens's Handbook on Testing Materials. (Henning.) Svo, 7 50 

Merriman's Text-book on the Mechanics of Materials 8vo, 4 00 

Strength of Materials 12010, 1 00 

Metcaif's Steel. A Manual for Steel-users i2mo, a 00 

Sabin's Industrial and Artistic Technology of Paints and Varnish. ...... 8vo, 3 00 

Smith's Materials of Machines iamo, 1 00 

Thurston's Materials of Engineering 3 vols., Svo, 8 00 

Part H.— Iron and Steel *. 8vo, 3 50 

Part HI? — A Treatise on Brasses, Bronzes, and Other Alloys and their 

Constituents 8vo a 50 

Text-book of the Materials of Construction 8vo, 5 00 

12 



Wood's Treatise on the Resistance of Materials and an Appendix on the 

Preservation of Timber 8vo, 2 00 

Elements of Analytical Mechanics 8vo, 3 00 

Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. . ,8vo, 4 00 



STEAM-ENGINES AND BOILERS. 

Carnot's Reflections on the Motive Power of Heat. (Thurston.) nmo, x 50 

Dawson's "Engineering" and Electric Traction Pocket-book. .t6mo, mor., 5 00 

Ford's Boiler Making for Boiler Makers i8mo, 1 00 

Goss's Locomotive Sparks 8vo, 2 00 

Hemenway's Indicator Practice and Steam-engine Economy i2mo, 2 00 

Hutton's Mechanical Engineering of Power Plants 8vo, 5 00 

Heat and Heat-engines 8vo, 5 00 

Kent's Steam-bo'ler Economy 8vo, 4 00 

Kneass's Practice and Theory of the Injector 8vo 1 50 

MacCord's Slide-valves 8vo, 2 00 

Meyer's Modern Locomotive Construction 4to, 10 00 

Peabody's Manual of the Steam-engine Indicator iamo, x 50 

Tables of the Properties of Saturated Steam and Other Vapors 8vo, x 00 

Thermodynamics of the Steam-engine and Other Heat-engines 8vo, 5 00 

Valve-gears for Steam-engines 8vo, 2 50 

Peabody and Miller's Steam-boilers 8vo, 4 00 

Pray's Twenty Years with the Indicator Large 8vo, 2 50 

Pupln's Thermodynamics of Reversible Cycles in Gases and Saturated Vapors. 

(Osterberg.) iamo, 1 25 

Reagan's Locomotives : Simple* Compound, and Electric 12210, 2 50 

Rontgen's Principles of Thermodynamics. (Du Bois.) 8vo, 5 00 

Sinclair's Locomotive Engine Running and Management 1 amo , 2 00 

Smart's Handbook of Engineering Laboratory Practice 12 mo, 2 50 

Snow's Steam-boiler Practice 8vo, 3 00 

Spangler's Valve-gears 8vo, 2 50 

Notes on Thermodynamics 1 2mo, x 00 

Spangler, Greene, and Marshall's Elements of Steam-engineering 8vo, 3 00 

Thurston's Handy Tables 8vo, x 50 

Manual of the Steam-engine 2 vols. 8vo , 10 00 

Part I. — History, Structuce, and Theory 8vo, 6 00 

Part H. — Design, Construction, and Operation 8vo, 6 00 

Handbook of Engine and Boiler Trials, and the Use of the Indicator and 

the Prony Brake 8vo 5 00 

Stationary Steam-engines 8vo, 2 50 

Steam-boiler Explosions in Theory and in Practice nmo x 50 

Manual of Steam-boiler? , Their Designs, Construction, and Operation . 8vo, 5 00 

Weisbach's Heat, Steam, and Steam-engines. (Du Bois.) 8vo, 5 00 

Whitham's Steam-engine Design 8vo, 5 00 

Wilson's Treatise on Steam-boilers. (Flather.) i6mo, 2 50 

Wood's Thermodynamics Heat Motors, and Refrigerating Machines. . . .8vo, 4 00 



MECHANICS AND MACHINERY. 

Barr's Kinematics of Machinery 8vo, 2 50 

Bovey's Strength of Materials and Theory of Structures 8vo, 7 50 

Chase's The Art of Pattern-making i2mo, 2 50 

ChordaL — Extracts from Letters nmo, 2 00 

Church's Mechanics of Engineering 8vo, 6 00 

Notes and Examples in Mechanics 8vo, 2 00 

13 



SANITARY SCIENCE. 

Copeland't Manual of Bacteriology. (In prtparation.) 

Folwell's Sewerage. (Designing, Construction and Maintenance,) 8vo, 3 00 

Water-supply Engineering 8vo, 4 00 

Fuertes's Water and Public Health iamo, 1 50 

Water-filtration Works nmo, 2 50 

Gerhard's Guide to Sanitary House-inspection i6mo, 1 00 

Goodrich's Economical Disposal of Town's Refuse Demy 8vo, 3 50 

Hazen's Filtration of Public Water-supplies 8vo, 3 00 

Kiersted's Sewage Disposal * iamo, x 25 

Leach's The Inspection and Analysis of Food with Special Reference to State 

Control. (In preparation.) 
Mason's Water-supply. (Considered Principally from a Sanitary Stand- 
point.) 3d Edition, Rewritten 8vo, 4 00 

Examination of Water. (Chemical and Bacteriological.) iamo, x 25 

Merriman's Elements of Sanitary Engineering 8vo, s •© 

Nichols's Water-supply. (Considered Mainly from a Chemical and Sanitary 

Standpoint.) (1883.) 8vo, 2 50 

Ogden's Sewer Design nmo, 2 00 

Prescott and Winslow's Elements of Water Bacteriology, with Special Reference 

to Sanitary Water Analysis nmo, 1 25 

* Price's Handbook on Sanitation iamo, x 50 

Richards's Cost of Food. A Study in Dietaries iamo, x 00 

Cost of Living as Modified by Sanitary Science i2mo, x 00 

Richards and Woodman's Air, Water, and Food from a Sanitary Stand- 
point 8ro, 2 00 

* Richards and Williams's The Dietary Computer 8vo, x 50 

Rideal's Sewage and Bacterial Purification of Sewage 8vo, 3 50 

Turneaure and Russell's Public Water-supplies 8vo, 5 00 

Whipple's Microscopy of Drinking-water 8vo, 3 50 

Woodhull's Notes and Military Hygiene x6mo, x 50 

MISCELLANEOUS. 

Barker's Deep-sea Soundings 8vo, 2 ©0 

Emmons's Geological Guide-book of the Rocky Mountain Excursion of the 

International Congress of Geologists Large 8vc 

Ferrel's Popular Treatise on the Winds 8vo 

Haines's American Railway Management X2mo> 

Mott's Composition, Digestibility, and Nutritive Value of Food. Mounted chart. 

Fallacy of the Present Theory of Sound x6mo 

Ricketts's History of Rensselaer Polytechnic Institute, 1824-1894. Small 8vo, 

Rotherham's Emphasized New Testament Large 8vo, 

Steel's Treatise on the Diseases of the Dog 8vo, 

Totten's Important Question in Metrology 8vo 

The World's Columbian Exposition ot 1893 4to, 

Von Behring'8 Suppression of Tuberculosis. (Bolduan.) (Inprett.) 
Worcester and Atkinson. Small Hospitals, Establishment and Maintenance, 
and Suggestions for Hospital Architecture, with Plans for a Small 
Hospital X2mo, x 25 

HEBREW AND CHALDEE TEXT-BOOKS. 

Green's Grammar of the Hebrew Language 8vo, 3 00 

Elementary Hebrew Grammar i2mo, x 25 

Hebrew Chrestomathy 8vo, 2 00 

Gesenius's Hebrew and Chaldee Lexicon to the Old Testament Scriptures. 

(TregeUes.) Small 4x0, half morocco. 5 00 

Letteris's Hebrew Bible .8vo, 2 2 

16 



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